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Maple User Manual - University Information & Technology Services

1. 4 isin A Maple set has the basic properties of a mathematical set e Each element is unique Repeated elements are stored only once e The order of elements is not stored 8 2 Creating and Using Data Structures 339 For example gt c a a a b c a a b c Using Sets To perform mathematical set operations use the set data structure gt 12 65 13 U 2 8 6 7 1 2 5 6 7 8 Note The union operator is available in 1 D Math input as union For more information refer to the union help page For more information on sets refer to the set help page Lists A list is an expression sequence enclosed in brackets gt b 2 3 3 1 0 L 2 3 31 90 Note Lists preserve both the order and repetition of elements Accessing Entries To refer to an element in a list e Use square brackets For example gt L 2 1 1 0 For more information see Accessing Elements page 338 Using Lists Some commands accept a list or set of expressions 340 8 Maple Expressions For example you can solve a list or set of equations using a context menu or the solve command gt solve x a y 0 x 2 y 2 x 1 y 1 For more information see Solving Equations and Inequations page 112 For more information on sets and lists refer to the set help page Arrays Conceptually the Array data structure 1s a generalized list Each element has an index that you can use to a
2. undefined Assumptions on Variables To simplify problem solving it is recommended that you always apply any known assump tions to variables You can impose assumptions using the assume command To apply as sumptions for a single computation use the assuming command Note The assume and assuming commands are not supported by the RealDomain package The assume Command You can use the assume command to set variable properties for example x real and re lationships between variables for example x lt 0 or x lt y For information on valid proper ties refer to the assume help page For information on the double colon operator refer to the type help page The assume command allows improved simplification of symbolic expressions especially multiple valued functions for example computing the square root To assume that x is a positive real number use the following calling sequence Then compute the square root of x gt assume O lt x y x The trailing tilde on the name x indicates that it carries assumptions When you use the assume command to place another assumption on x all previous assump tions are removed gt assume x lt 0 y a Displaying Assumptions To view the assumptions on an expression use the about com mand 4 6 Restricting the Domain 145 gt about x Originally x renamed x is assumed to be RealRange infinity Open 0 Imposing Multiple Assumptions To
3. foreground onclick refer rice tooltip visible clickButton1 true 10 5 Authoring Maplets 407 10 In the Evaluate Expression dialog that displays the Target drop down list con tains the defined elements to which you can send information in this case Plot terl and TextField1 The List group box located below the Expression group box displays the defined elements to which gt Expression you can retrieve information in this case lot TextFieldl x 10 10 TextField1 Argument Forti Command Form rA a In the Target drop down list select p Plotter1 e b In the Command Form tab enter plot TextField1 x 10 10 in the Expres sion group box Note Do not include a semicolon at the end of the plot com mand You can also double click Text Field1 in the List group box to insert this element in the command syntax c Click Ok Run the Maplet 11 From the File menu select Run You are prompted to save the Maplet Maplets created with the Maplet Builder are saved as maplet files 12 Click Yes and navigate to a location to save this Maplet For further information on the Maplet Builder see the 7MapletBuilder help page For more examples of designing Maplets using the Maplet Builder see MapletBuilder ex amples Maplets Package When designing a complicated Maplet the Maplets package offers greater control The Maplets Elem
4. L7 TTTTITTS To convert absolute temperatures the Unit Converter uses the convert temperature command For example 32 degrees Fahrenheit corresponds to 0 degrees Celsius gt convert 32 temperature degF degC 0 Applying Units to an Expression To insert a unit use the Units palettes The Units FPS palette Figure 4 7 contains im portant units from the foot pound second system of units The Units SI palette Figure 4 8 contains important units from the international system of units W Units FPS W Units 57 unt LA unt em Is ls Lpowrndea INI xe Pal 1 PI Vl K F T41 F ieh Ic 2 I HP peundal LE red sr rad sr wel ix Lie LS We Mp Figure 4 7 Units FPS Palette Figure 4 8 Units SI Palette To insert a unit e Ina Units palette click a unit symbol gt 3i 3 Lf To insert a unit that is unavailable in the palettes 1 In a Units palette click the unit symbol EA Maple inserts a Unit object with the placeholder selected 132 4 Basic Computations 2 In the placeholder enter the unit name or symbol For example to enter 0 01 standard the default context miles you can specify the unit name mile or symbol mi gt 0 01 mile 0 01 mi The context of a unit is displayed only if it 1s not the default context Important In 1 D Math input the quantity and unit entered using the top level Unit comm
5. x 7 3 x 1 4 x 4 x 7 3 x 1 4 x 4 expand _ 6x gt 24x 15 y 24x 18 5 8 Clickable Math Examples 223 Change the X and y axis ranges using context menus 7 By default plots generated using the context menus have an X axis range of 10 to 10 To change the range right click the plot and select Axes Properties In the Horizontal tab of the Axes Properties dialog de select Use data ex tents and change the Range min and Range max to 0 and 5 respectively Click the Vertical tab and de select Use data ex tents Change the Range min and Range max to 5 and 10 respectively 8 Click OK to apply the changes and return to the plot The interception points of this graph with the X axis are 1 and 3 the same solutions that we found previously Example 3 Solve a Quadratic Trig Equation Find all of the solutions to the equation 6 cos x cos x 2 0 in the interval 0 27 We solve this problem using the following methods e Graphical Solution page 223 Solution by Task Template page 225 e Analytic Solution page 225 Graphical Solution 1 Ctrl drag the equation 6 cos x cos x 2 0 to a blank docu ment block and press Enter 2 Right click the output and select Left hand Side 224 e 5 Mathematical Problem Solving 3 Right click the output and select Plots Plot W Interactive Plot
6. 0 0913252028230576718 x 10 9041216700744900 gt with Optimization InmportMPS Interactive LPSolve LSSolve Maximize Minimize NLPSolve OPSolve gt NLPSolve sm y l 15 0 0913252028230576718 x 10 9041216700744900 For more information on optimization see Optimization page 188 To unload a package e Use the unwith command specifying the package as an argument gt unwith Optimization Alternatively use the restart command The restart command clears Maple s internal memory The effects include unassigning all names and unloading all packages For more information refer to the restart help page Note To execute the examples in this manual you may be required to use the unassign or restart command between examples Some packages contain commands that have the same name as a top level command For example the plots package contains a changecoords command Maple also contains a top level changecoords command gt with plots 86 3 Worksheet Mode After the plots package is loaded the name changecoords refers to the plots changecoords command To use the top level changecoords command unload the package or use the restart command For alternative methods of accessing the top level command see the rebound help page Top Packages Here are a few of the most frequently used Maple packages A complete list of packages is available in the Maple help system at Help Ma
7. Canvas Style You can alter the Canvas in the following ways e Add a grid of horizontal and or vertical lines By default the canvas opens with a grid of horizontal and vertical lines e Change the grid line color e Change the spacing between grid lines e Change the background color These options can be changed in the Drawing Properties Canvas Icon See Figure 7 17 322 e 7 Creating Mathematical Documents Emma Horizontal vertical Canvas Figure 7 17 Drawing Properties Canvas Icon Change the Gridline Color Inserting Images You can insert images in these file formats into your document e Graphics Interchange Format gif e Joint Photographic Experts Group jpe jpeg jpg Portable Network Graphics png Bitmap Graphics bmp Tagged Image File Format tif tiff Jfx Portable aNyMap pnm Kodak FlashPix fpx To insert an image into the document at the cursor location 1 From the Insert menu select Image The Load Image dialog opens 2 Specify a path or folder name 3 Select a filename 4 Click Open The image is displayed in the document If the source file is altered the embedded image does not change because the original object is pasted into the document 7 6 Hyperlinks 323 To resize an inserted image 1 Click the image Resizing anchors appear at the sides and corners of the image 2 Move the mouse over the resize
8. For further examples using both the MapletBuilder and Maplets package commands see the Maplets example worksheets For a listing refer to the examples index help page 10 5 Authoring Maplets 409 Saving When saving a Maplet you can save the document as an mw file or you can export the document as a maplet file Maple Document To save the Maplet code as an mw file 1 From the File menu select Save 2 Navigate to the save location 3 Enter a filename 4 Click Save If the document contains only Maplet code it is recommended that you export the document as a maplet file Maplet File To export the Maplet code as a maplet file 1 From the File menu select Export As 2 In the Files of Type drop down list select Maplet 3 Navigate to the export location 4 Enter the filename 5 Click Save 410 10 Embedded Components and Maplets 11 Input Output and Interacting with Other Products 11 1 In This Chapter Writing to Files page 411 Saving to Maple Saving Data to a File file formats Saving Expressions to a File Reading from Files page 414 Opening Maple Reading Data from a File files Reading Expressions from a File Exporting to Other Formats page 416 Export Exporting Documents ing documents in file formats supported by MapleNet other software Maple T A Connectivity page 420 Using Maple with Translating Maple Code to Other Programming other programming languages and softwa
9. LaTeX The tex file generated by Maple is ready for processing by LaTeX All distributions of Maple include the necessary style files By default the LaTeX style files are set for printing the tex file using the dvips printer driver You can change this behavior by specifying an option to the usepackage LaTeX command in the preamble of your tex file For more in formation refer to the exporttoLaTeX help page 11 4 Exporting to Other Formats e 417 Maple Input You can export a Maple document as Maple input so that it can be loaded using the Maple Command line version Important When exporting a document as Maple input for use in Command line Maple your document must contain explicit semicolons in 1 D Math input If not the exported mpl file does not contain semicolons and Command line Maple generates errors Maplet Application The Export as Maplet facility saves a Maple document as a maplet file so that you can run it using the command line interface or the MapletViewer The MapletViewer is an executable program that can launch saved Maplet applications It displays and runs Maplet applications independently of the Maple Worksheet interface Important When exporting a document as a Maplet Application for use in Command line Maple or the MapletViewer your document must contain explicit semicolons If not the exported maplet file does not contain semicolons and Command line Maple and the MapletViewer generate
10. 7 Creating Mathematical Documents Security Levels By default Maple prompts the user before automatically executing the document To set security levels for the autoexecute feature use the Security tab in the Options dialog For details refer to the 0ptionsDialogSecurity help page 7 4 Tables Tables allow you to organize content in a document Creating a Table To create a table 1 From the Insert menu select Table 2 Specify the number of rows and columns in the table creation dialog 3 Click OK The default properties for the table include visible borders and auto adjustment to 100 of the document width These options as well as the table dimensions can be modified after table creation Create a table with 4 rows and 2 columns at the end of your document In document mode the input mode is set to Math by default in worksheet mode the default is Text mode Cell Contents Any content that can be placed into a document can also be placed into a table cell including other sections and tables Table cells can contain a mix of e Input commands e 2 D Math e Embedded components buttons sliders check boxes and more e Plots e Images 7 4 Tables 307 Enter a heading in both columns of the first row in 2 D Math You can use any text formatting features within each cell for example bold and center the headings Navigating Table Cells Use the Tab key to move to the next cell Ensure that the T
11. Assist ants CurveFitting which fits and plots a curve through the points For more information refer to the CurveFitting Interactive help page 258 6 Plots and Animations gt pointplot 0 1 1 1 3 0 4 3 2 0 4 1 3 2 4 1 axes BOXED symbolsize 25 symbol circle The matrixplot Command The matrixplot command plots the values of a plot object of type Matrix The matrixplot command accepts options such as heights and gap to control the appearance of the plot For more information on Matrices see Linear Algebra page 159 gt with LinearAlgebra 6 2 Creating Plots 259 gt A HilbertMatrix 6 pirt i 23 4 5 6 14d 4 i 2 34 5 6 7 Lita tit 2 3 4 6 68 7 8 A gt Lididdai i a3 6 7 8 9 LJE LAA gt G6 7 9 W L LT l gt B i ToeplitzMatrix 1 2 3 4 5 6 symmetric 234 123 252 a2 4321 654321 e Un on mS Ww A 2 3 4 to Ww i 260 6 Plots and Animations gt matrixplot A B heights histogram axes normal gap 0 25 style patch The contourplot Command The contourplot command generates a topographical map for an expression or function To create a smoother and more precise plot increase the number of points using the num points option 6 2 Creating Plots 261 gt contourplot cos x y x 4 4 y 4 4 filled true numpoints 750 262 6 Plots and Animations Multiple Plots in the Same Plot
12. b Click OK to proceed to the Select Plot Type window Plot the expression 4 In the Select Plot Type window notice the default setting of a 2 D plot type and an x axis range 10 10 Notice also the various plot types available for this expression 5 Click Plot To see the Maple syntax used to generate this plot see Maple commands from Creating Plots Interactive Plot Builder page 250 242 e e 6 Plots and Animations Example 2 Display a plot of multiple expressions in 1 variable Maple can display multiple expressions in the same plot region to compare and contrast The Interactive Plot Builder accepts multiple expressions Launch the Interactive Plot Builder and enter the expressions 1 Launch the Interactive Plot Builder The Plot Builder accepts expressions in 1 D Math and performs basic calculations on expressions For example entering diff sin x 2 x in the Specify Expression window performs the calculation and displays the expression as 2 cos x 2 x in the Expression group box 2 In the Specify Expressions window a In three separate steps add the expressions sin x 2 diff sin x 2 x and int sin x 2 x Change the x axis range 3 In the Select Plot Type window a Change the x Axis range to 3 3 b Click Options to proceed to the Plot Options window Launch the Plot Options window and return the plot command syntax to the document 4 Click Command Display the actual plot 5 Execute the in
13. gt LinearSolve basis lt 25 4 9 gt 170 9 285 9 786 9 Numeric Computations You can very efficiently perform computations on large matrices and vectors that contain floating point data using the built in library of numeric linear algebra routines Some of these routines are provided by the Numerical Algorithms Group NAG Maple also contains portions of the CLAPACK and optimized ATLAS libraries For information on performing efficient numeric computations using the LinearAlgebra package refer to the EfficientLinearAlgebra help page See also Creating Matrices and Vectors with Specific Properties page 164 and Reading from Files page 414 Student LinearAlgebra Package The Student package contains subpackages that help instructors teach concepts and allow students to visualize and explore ideas These subpackages also contain computational commands In the Student LinearAlgebra subpackage the environment differs from that of the Lin earAlgebra package in that floating point computations are generally performed using 5 4 Calculus 175 software precision instead of hardware precision and symbols are generally assumed to represent real rather than complex quantities These defaults and others can be controlled using the SetDefault command For more information refer to the Student LinearAl gebral SetDefault help page For information on using Maple as a teaching and learning tool see
14. 2 735838887 10 x 7 985305551 10 x 7 52701389 10x 0 044193661 18x L 0 O8838747155x 0 0441940000 Descriptions of Assistants The remaining assistants are described below Some of the assistants are interfaces to package commands For more information on package commands see Package Com mands page 47 e Back Solver an interface that allows you to take a mathematical formula involving multiple parameters enter values for all but one of the parameters and solve for the re maining value You can also plot the behavior of the formula as one of the parameters change e Curve Fitting an interface to commands in the CurveFitting package Data points can be entered as independent and dependent values and interpolated with polynomials ra tional functions or splines e Data Analysis an interface to the data analysis commands in the Statistics package e Equation Manipulator an interface for interactively performing a sequence of operations on an equation You can group terms apply an operation to both sides of the equation complete the square and so on e Import Data an interface to read data from an external file into Maple 1 3 Point and Click Interaction 37 Installer Builder an interface to the InstallerBuilder package in which you can create installers for your Maple toolboxes For information on toolboxes go to http www maplesoft com developers index aspx Library Browser an interfa
15. 9 4 Procedures A Maple procedure is a program consisting of Maple statements Using procedures you can quickly execute the contained sequence of statements Defining and Running Simple Procedures To define a procedure enclose a sequence of statements between proc and end proc statements In general you assign a procedure definition to a name The following procedure returns the square root of 2 384 9 Basic Programming gt p proc sqrt 2 end proc p proce sqrt 2 end proc Note Maple returns the procedure definition To improve readability of procedures it is recommended that you define a procedure using multiple lines and indent the lines using space characters To begin a new line without evaluating the incomplete procedure definition press Shift Enter When you have finished entering the procedure press Enter to create the procedure For example gt p proc sqrt 2 end proc To run the procedure p enter its name followed by parentheses gt pl V2 Procedures with Inputs You can define a procedure that accepts user input In the parentheses of the proc statement specify the parameter names For multiple parameters separate the names with commas gt geometric_mean proc x y sqrt x y end proc When the user runs the procedure the parameter names are replaced by the argument values gt geometric_mean 13 17 221 gt geometric_mean 13
16. Lists of Common Commands and Packages Palettes page amp 7 Items that you can insert by Using Palettes clicking or dragging Context Menus page 89 Pop up menus of e Using Context Menus common operations 79 SO 3 Worksheet Mode Assistants and Tutors page 91 Graphical inter Launching Assistants and Tutors faces with buttons and sliders Task Templates page 91 Sets of commands Viewing Task Templates with placeholders that you can insert and use to fi task Inserting a Task Template perform a tas Performing the Task Text Regions page 93 Areas in the document Inserting a Text Region in which you can enter text Formatting Text Names page 94 References to the expressions e Assigning to Names you assign to them Unassigning Names Valid Names Equation Labels page 97 Automatically gener Displaying Equation Labels ated labels that you can use to refer to expressions Referring to a Previous Result Execution Groups with Multiple Outputs Label Numbering Schemes Features of Equation Labels 3 2 Input Prompt In Worksheet mode you enter input at the Maple input prompt gt The default mode for input is Math mode 2 D Math To evaluate input e Press Enter Maple displays the result output below the input 3 For example to find the value of sin l enter the expression and then press Enter 3 gt in wa oo ue ja 3 1 For example compute th
17. Numerically Solve From point hs ia Obtain Solutions For Integral Transforms ae Solve explicit 2 D Math F Solve general solution Solve For Variable b Figure 2 3 Finding the Approximate Solution to an Equation 72 x solve 3 1 14 1120 3 1 14 1120 S12 gt of x Of x SC IMa 3 m 14 ie F 14 we Ix lv os 12 2 200603126 2 337021648 It 2 6 Performing Computations 75 For more information on solving equations including solving inequations differential equations and other types of equations see Solving Equations page 111 Using Units You can create expressions with units To specify a unit for an expression use the Units palettes The Units FPS palette Figure 2 4 contains important units from the foot pound second FPS system of units used in the United States The Units SI palette Figure 2 5 contains important units from the international system SI of units JW Units FPS OW Units 51 et LA unt Ie Isl ls Lpowrndea INI xg Pal 1b WI VI KI poundfarce F 14 F inch Ic e I HP poundal j rad sr red sr eeil bxi Le ISI We P Figure 2 4 FPS Units Palette Figure 2 5 SI Units Palette To insert an expression with a unit 1 Enter the expression 2 In a unit palette click a unit symbol Note To include a reciprocal unit divide by the unit To evaluate an expression that contains units 1 Enter the expr
18. Plots Sorts Units 2 D Math FTF F F F FT F F i F F F F r 2 D Plot 3 D Plot D Implicit Plot gt 3 D Implicit Plot gt Plot Builder 1 2 Entering Expressions 19 Result 2x 9 LQ 10 5 0 5 10 X 10 20 Saving a Maple Document To save these examples you created from the File menu select Save Maple documents are saved as mw files 1 2 Entering Expressions Execution Groups An execution group is a grouping of Maple input with its corresponding Maple output It is distinguished by a large square bracket called a group boundary at the left An execution group may also contain any or all of the following a plot a spreadsheet text embedded components and a drawing canvas Execution groups are the fundamental computation and documentation elements in the document If you place the cursor in an input command and press the Enter or Return key Maple executes all of the input commands in the current execution group 20 e 1 Getting Started Math Mode vs Text Mode The default mode of entry in Document or Worksheet mode is Math Mode which displays input in 2 D Math In earlier releases of Maple commands and expressions were entered using Maple Input or 1 D Math Important With Maple input you must terminate commands with a semicolon or colon gt cos alpha 2 sin alpha 2 cos a sin a gt a int exp sqrt 2 x x S re co Pi Be g
19. des Revert 7 Now that the integral has been evaluated click Close to close the tutor and return the evaluated integral to the document integration methods tutor ema E Solution by First Principles l 1 Ctrl drag the integrand toa y 4 x blank document block region and press Enter Perform trig substitution evaluate at point HH 2 Right click the output and select Evaluate at a point In the dialog that displays enter 2 sin u 5 i 4 4sin u 3 Right click the output and select Simplify simplify symbolic _ Symbolic l 2 cos 2 5 8 Clickable Math Examples 233 du x 2sin u Calculate alculate dx x 2sin u 4 In a blank document block enter the substitu tion equation x 2sin u and press Enter implicit differentiation 5 Right click the output and select Differentiate Implicitly In the dialog that displays change the Independent Variable to u 2 cos u Calculate the integral in terms of H 5 9 5 10 6 Referencing the results by their equation labels multiply the original simplified expression by this derivative 7 Integrate the resulting expression s 11 du Revert the substitution x 2sin 5 12 8 Place the equation x 2sin u ina blank x 2sin w document block Delete and insert the equation label for the previous result the value of the integ _Sve for u ral in terms of t Press Enter
20. none selected Current Position ie Spacing of Major Tick Marks Spacing of Minor Tick Marks Action When Value Changes bgi Options Enable Input Use Specified Text Width Visible Show Track Orient Vertically LC Show Axis Labels Show Axis Tick Marks Snap to Axis Tick Marks Update Continuously while Dragging Figure 10 2 Label Properties Dialog Figure 10 3 Slider Properties Dialog Options Enabled 5 Name the component SliderLabel and click Ok 6 Right click Control click Macintosh the slider component Select Component Prop erties The Slider Properties dialog opens See Figure 10 3 7 Name the component Slider1 8 Enter the value at the lowest position as 0 and the highest as 100 9 Enter major tick marks at 20 and minor tick marks at 10 10 To define an action click the Edit button for the Action When Value Changes The dialog that opens allows you to program the action of displaying the slider value in the label component The dialog includes instructions on how to program embedded components The use in end use statement allows you to specify routines using the short form of ac cessing a package command without invoking the package For details on this command refer to the use help page 11 Before the end use statement at the bottom of the dialog enter the following command Do SliderLabel caption 0Slider1 value 12 Click OK 13 Make sure that the Update Con
21. xcos ax a a Recall that you can also enter symbols including and d using symbol completion e Enter the symbol name or part of the name for example int or d and then press the completion shortcut key For more information see Symbol Names page 29 You can also compute an indefinite integral using context menus For more information see Context Menus page 39 To compute the definite integral of an expression ve 1 In the Expression palette click the definite integration item a 2 Specify the endpoints of the interval of integration integrand expression and variable of integration and then evaluate it For example to integrate e In t over the interval 0 gt e In t de 0 f e In t Ei 1 at y lnla lim a Maple treats the parameter a as a complex number As described in Assumptions on Vari ables page 144 you can compute under the assumption that a is a positive real number using the assuming command gt e In t dtassuminga gt 0 0 y In a al 5 4 Calculus 185 To compute iterated integrals line integrals and surface integrals use the task templates Tools Tasks Browse in the Multivariate and Vector Calculus folders The int Command b fd and f dx use the int command To use the int command directly specify the fol cf lowing arguments e Expression to integrate e Variable of integ
22. 134 extensibility 139 objects 137 ScientificErrorAnalysis package description 86 ScientificErrorAnalysis package 139 extensibility 142 objects 140 search help system 56 sections in worksheet 295 security levels auto execute 306 security tab options dialog 306 select command 381 438 e Index selection execute 10 selectremove command 381 semicolon 81 82 seq command 380 series 182 command 182 plotting 183 Taylor 182 type 183 sets 338 shape option 165 show worksheet content 298 show contents dialog using 298 significant digits 104 simplify command 353 360 sketch pad canvas style 299 slider embedding 329 Slider component 391 solutions assigning as expression 119 assigning as function 120 details 125 formal 125 formal power series 125 integers 126 real 143 series 125 verifying 119 solve equations 112 for real solutions 143 numerically 117 symbolically 114 inequations 112 for real solutions 143 symbolically 114 integer equations 126 linear system 126 173 modular integer equations 126 ODEs 121 PDEs 125 recurrence relation 127 transcendental equations 116 solve command 114 340 finding all solutions 116 finding parametric solutions 117 real solutions 143 solving procedures 117 sort lists 358 polynomials 152 358 sort command 152 358 plex option 153 spacing format 286 Special Functions Assistant 37 spellcheck 332 American spelling
23. 2 498755763 Complex Solutions To search for a complex solution or find all complex and real roots for a univariate polynomial specify the complex option for the fsolve command gt fsolve equation3 y complex 1 13846246879373 0 485062494059435 I 1 13846246879373 0 485062494059435 1 0 336532273926790 1 94039266366067 If the fsolve command does not find any solutions it is recommended that you specify a range in which to search for solutions or specify an initial value Range To search for a solution in a range specify the range in the calling sequence The range can be real or complex gt fsolve equation2 z z 100 200 149 2390528 The syntax for specifying a region in the complex plane is lower left point upper right point gt fsolve equation3 y y 2 I 0 complex 1 13846246879373 0 485062494059435 Initial Values You can specify a value for each unknown The fsolve command uses these as initial values for the unknowns in the numerical method 4 4 Solving Equations 119 gt fsolve equation2 z 100 z 98 98037599 4 9 For more information and examples refer to the fsolve details help page For information on verifying and using solutions returned by the fsolve command see the following section Working with Solutions Working with Solutions Verifying It is recommended that you always verify solutions that the solve and fsolve commands return us
24. 204 numeric 185 surface 185 with units 133 interactive commands Student 38 Interactive Linear System Solving tutor 76 Interactive Plot Builder Assistant creating animations 270 creating plots 238 customizing animations 276 customizing plots 264 interface command rtablesize option 164 verboseproc option 386 international system SI 129 InterquartileRange command 194 interval arithmetic 139 iquo command 108 iroot command 108 is command 145 isprime command 108 isqrt command 108 italic format 284 J J entering 111 Jordan form 171 K keyboard keys Command Completion xiv Context Menu xiv keystrokes 7 L Label component 390 labels 99 last name evaluation 366 lcm command 158 lcoeff command 156 Idegree command 157 least squares 173 left single quotes 96 left hand side 350 levels of evaluation 365 lexicographic order 153 Ihs command 350 Library Browser description 37 Limit command 176 limit command 175 limits 175 multidimensional 176 line break 286 line integrals 204 linear algebra 173 computations 168 efficiency 164 174 LinearAlgebra package 173 teaching 174 199 Linear System Solving tutor 76 linear systems solving 126 173 interactive 76 LinearAlgebra package description 86 LinearAlgebra package 171 commands 173 numeric computations 174 LinearSolve command 126 List Box component 390 lists 168 339 returning solutions as 115 local variable
25. 271244 187092 e 613868 You can specify scalar multiplication explicitly by entering which displays in 2 D Math as In 2 D Math you can also implicitly multiply a scalar and a matrix or vector by placing a space character between them In some cases the space character is optional For example Maple interprets a number followed by a name as an implicit multiplication In 2 D Math exponents display as superscripts A few additional matrix and vector operators are listed in Table 5 6 Define two column vectors gt d lt 1 2 3 gt e lt 4 5 6 gt 5 3 Linear Algebra 171 Table 5 6 Select Matrix and Vector Operators o Hermitian Transpose ny H _ oH I 21 gt a 4l 2 1 Cross Product amp X gt with LinearAlgebra 3 D vectors only gt d amp xe Exponential operators display in 2 D Math as superscripts After loading the LinearAlgebra package the cross product operator is available as the infix operator amp x Otherwise it is available as the LinearAlgebra CrossProduct command For information on matrix arithmetic over finite rings and fields refer to the mod help page Point and Click Interaction Using context menus you can perform many matrix and vector operations Matrix operations available in the context menu include the following e Perform standard operations determinant inverse norm 1 Euclidean infinity or Frobenius transpose and trace e C
26. 77 3 Right click the matrix and select Tu tors Linear Algebra Linear Sys Linear Algebra Linear System Solving tem Solving The Linear System Solving dialog appears where you can choose the solving method Gaussian Elimination reduces the matrix to row echelon form then performs back substi tution to solve the system Gauss Jordan Elimination reduces the matrix to re duced row echelon form where the equations are already solved For this example choose Gaussian Elimination Gaussian Elimination Gauss Jordan Elimination 4 The Gaussian Elimination dialog FI paren peer cree rn eterna opens You can specify the Gaussian File Edt Help elimination step by step or you can use s 67 3 4 ee ee the Next Step or All Steps buttons to 4 s 34 in 2 Click on any button to have Maple perform the steps for you 4o 9 23 10 Perea Wee eee Add multiple oe 4 2 17 434 5 Once the matrix is in row echelon upper triangular form click the Solve System button to move to the next step Multiply Swap Edit Matrix i Solve System 78 2 Document Mode 6 The Solve the system of equations in Solve the RES RE aE Rn Ae Row Echelon Form dialog appears cn nc n Click the buttons on the right to calculate Hna System of Equations the solution first find the Equations then solve for each variable Click the Solution button to display the solution
27. Click Plot to open the Interactive Parameter window Note To apply plot options before interactively adjusting the plot click Options to open the Plot Options window After setting the plot options click Plot to display the Interactive Parameter window 6 To adjust the numeric values use the slider 7 Click Done to place the plot in the Maple document To see the Maple syntax used to generate this plot see Maple commands from Creating Plots Interactive Plot Builder page 250 For information on customizing plots using the Interactive Plot Builder refer to Custom izing Plots Interactive Plot Builder Options page 264 Context Menu A context menu in Maple displays a list of commands to manipulate display or calculate using a Maple expression The commands in the menu depend on the type of the expression To display the context menu for a Maple expression right click Control click for Macin tosh the expression For expressions the context menu lists e 2 D or 3 D plot e 2 D or 3 D implicit plot e Interactive Plot Builder based on the expression selected When you invoke the Interactive Plot Builder through the context menu the expression automatically passes to the builder and Maple does not display the Specify Expression window One advantage of using the context menu is the simplicity of creating an expression using menus By using this method you do not need any knowledge of plot command syntax 6
28. Constants and Symbols Punctuation insert punctuation symbols such as inserting the re gistered trademark and copyright symbols into text regions Miscellaneous insert miscellaneous math and other symbols outside the above categories H si uh R Yc Alphabetical Palettes Greek Script A Fraktur W Open Face iB Cyrillic IK Dia Greek critical Marks Roman Extended Upper Case Roman Ex Go tended Lower Case D egraweoreer me 1 0a ZTP A A X drt R TE Ge GO JAD SBD amp m H U gt u mo O A D Viewing and Arranging Palettes By default palettes display in palette docks at the right and left sides of the Maple window To view and manage palettes and palette docks see Table 1 8 1 2 Entering Expressions 25 Table 1 8 Managing Palettes To view palette docks View Insert Format From the View menu select Palettes and then Tanpa PrF Expand Docks There are docks on the far Y S0ntext Bar PE aeee right and left of the window w Status Bar xt w Markers Task Elements 2D Math sik Slideshow Palettes rrange Palettes zoom Factor Show Palette Typesetting Rules Show All Palettes Show Hide Contents Show Default Palettes Header Footer Expand All Palettes Collapse All Palettes Expand Docks hs Expand All Sections Collapse Docks Collapse All Sections 26 1 Getting Started To add a palette Remove Palette 1 Rig
29. For example to display a plot in your document without saving the plot making your doc ument use less memory you can set a plot command to autoexecute 7 3 Commands in Documents 305 1 After the plot instruction enter a Maple prompt Insert Execution Group After Cursor 2 Enter the plot command plot sin x fsin dr and press Enter to execute 3 Select the plot then select Edit Remove Output From Selection 4 Place the cursor in the plot command the select Format Autoexecute Set 5 Save and close the document on reopening the command 1s re executed Result plot create a two dimensional plot Calling Sequence plot f x plotif x x0 x1 plot w1 2 Parameters f expression in independent variable x F independent variable 0 x1 le and nght endpoints of horizontal range Wil w2 x coordinates and y coordinates Plot the expression sin x and its derivative a sin x cos x in the same plot fl dx i gt plat ID gt zef Removing the Auto Execute Setting W een dx To remove the setting in a region 1 Select the region 2 From the Format menu select Autoexecute and then Clear To remove all autoexecuted regions from a document From the Format menu select Autoexecute and then Clear All Repeating Auto Execution To execute all marked groups e From the Edit menu select Execute and then Repeat Autoexecution 306
30. If equation label display is turned off complete both of the following operations e From the Format menu select Equation Labels and then ensure that Worksheet is selected e Inthe Options dialog Tools Options on the Display tab ensure that Show equation labels is selected Referring to a Previous Result Instead of re entering previous results in computations you can use equation label references Each time you need to refer to a previous result insert an equation label reference To insert an equation label reference e From the Insert menu select Label Alternatively press Ctrl L Command L Macintosh e In the Insert Label dialog see Figure 3 5 enter the label value and then click OK 98 3 Worksheet Mode Insert Label Type Equation Identifier Figure 3 5 Insert Label Dialog Maple inserts the reference For example To integrate the product of 3 4 and 3 5 Result in Document gt Tax 1 In the Expression palette click the indefinite integ cle ration item f The item is inserted and the integ rand placeholder is highlighted 2 Press CtrI L Command L for Macintosh 3 In the Insert Label dialog enter 3 4 Click OK Insert Label Type Equation haa Identifier 3 4 4 Press 5 Press CtrlI L Command L for Macintosh 6 In the Insert Label dialog enter 3 4 Click OK 7 To move to the variable of integration placeholder press Tab 8
31. Table 5 1 Polynomial Arithmetic Operators a lt dasencurrssncaveeiosasadenedenmionanbihetesedins 151 Table 5 2 Polynomial Coefficient and Degree Commands ccccceceeeee enone 156 Table 5 3 Select Other Polynomial Commands 20 lt ccsesuciacserseaonaapeaecenssanaaseseccoann 157 Table 5 4 Additional Polynomial Help 0 0 0 0 ccc cece cece ce ne ea eee eeceeeaeeneeseneeas 158 Table 5 5 Matrix and Vector Arithmetic Operators ccc ccc ccec ene eneeeeeeenseneeas 169 Table 5 6 Select Matrix and Vector Operators scicsicecanssrsccaactassanssacacratesssacceeeeesades 171 Table 5 7 Select LinearAlgebra Package Commands ciccccvsicsssanicssunavecsicsiceatenses 173 TOES ets T a E EE EEA E E EE AE AE EE EEE 176 Table 5 9 Optimization Package Commands ccccccceceneec ees eeteeseaeeseeseneeas 192 Table 5 10 Student and Instructor Resources cece ccc e cece eee cence eee eeaeeseneenees 198 Table 6 1 Windows of the Interactive Plot Builder cc cece cece cee eneeneeeeeees 239 Table 6 2 The plot and plot3d Commands c cece cece eee ee ee eeceeeeeeneeaeenenees 250 Table 6 3 Common Plot Options s csactesateseancceacwionauconsGnadaecaniancieeemecosantaaiexsteas 267 Table 6 4 Plot Analysis Options 3 cect cncacccaessdsaxtcacancecadeess dadvendedeueessemmaedeaeteciens 269 Table gt Vie animate Command 422eccssiccee dete casnnecueesanecen lt overedatsuaseeaeausaacennieceess 2
32. When dragging an expression to a plot region you can either make a copy of the expression from the input region or you can cut the expression thereby removing it from the input region To make a copy of the expression select the full expression in the input region and press Ctrl Command Macintosh while you drag the expression to the plot region To cut the expression and paste it in the plot region highlight the expression and drag it to the plot region 4 Repeat steps 2 and 3 using the following expressions sin 2x sin x 2 and sin x 5 To remove an expression from the plot region drag and drop the expression plot from the plot region to a Maple input region NN The plot and plot3d Commands The final method for creating plots is entering plotting commands The main advantages of using plotting commands are the availability of all Maple plot structures and the greater control over the plot output Plot options are discussed in Custom izing Plots page 264 250 6 Plots and Animations Table 6 2 The plot and plot3d Commands plot plotexpression x a b plot3d plotexpression x a b y a b e plotexpression expression to be plotted e x a b name and horizontal range e y a b name and vertical range Maple commands from Creating Plots Interactive Plot Builder The following examples show the plotting commands returned by the examples in Interactive Plot Builder page 238 Example
33. Y e gt colle 2axy c y z3 az 13 by 3 3 2 z gt y F 2ax tex I3 b y az x Coefficients and Degrees Maple has several commands that return coefficient and degree values for a polynomial See Table 5 2 Table 5 2 Polynomial Coefficient and Degree Commands coeff Coefficient of specified degree term i a gt coe aos Leading coefficient tcoeff Trailing coefficient i 4 gt coeff ae 25 J 5 5 2 Algebra 157 coeffs Sequence of all coefficients in one to one correspondence with the terms Note It does not return zero coefficients Highest degree Lowest degree term with a non zero coeffi cient Factorization To express a polynomial in fully factored form use the factor command gt factor x 1 x 1 x 1 f 1 The factor command factors the polynomial over the ring implied by the coefficients for example integers You can specify an algebraic number field over which to factor the polynomial For more information refer to the factor help page The ifactor command factors an integer For more information see Integer Operations page 106 To solve for the roots of a polynomial use the solve command For information on the solve command see Solving Equations and Inequations page 112 The isolve command solves an equation for integer solutions For more information see Integer Equations page 126 Other Commands Table 5 3 lists other commands avail
34. as you would do if you were per forming the calculation by hand and not to numeric approximations as you normally obtain from a standard hand held calculator gt sin 1 sin z sin x sin 1 0 sin x gt francs di In cos 32 4J 2 Floating Point Computations In some situations a numeric approximation of an exact quantity is required For example the plot command requires the expression it is plotting to evaluate to numeric values that can be rendered on the screen a cannot be so rendered but 3 14159 can be Maple distin 104 4 Basic Computations guishes approximate from exact quantities by the presence or absence of a decimal point 1 9 is approximate while is exact Note An alternative representation of floating point numbers called e notation may not include an explicit decimal point Je5 100000 3e 2 03 In the presence of a floating point approximate quantity in an expression Maple generally computes using numeric approximations Arithmetic involving mixed exact and floating point quantities results in a floating point result P gt lots r3 2 166666667 If a mathematical function is passed a floating point argument it normally attempts to produce a floating point approximation of the result 1 0 gt sin 1 5 e dx 0 0 0 9974949866 1 718281828 Converting Exact Quantities to Floating Point Values To convert an exact quantity to a numeric approximati
35. automatically grades responses as students complete assignments and tests For more information visit http www maplesoft com mapleta 420 11 Input Output and Interacting with Other Products Exporting Assignments to Maple T A You can use Maple to create graded questions for use in Maple T A For information on creating and testing questions see Creating Graded Assignments page 334 Using the Maple T A export feature you can create and test Maple T A content To export the document 1 From the File menu select Export As 2 In the Export As dialog specify a filename and the Maple T A zip file type The zip file containing your questions and assignment can be uploaded to Maple T A as a course module Any document content outside Maple T A sections indicated by green section markers is ignored by the export process For more details refer to the exporttoMapleTA help page 11 5 Connectivity Translating Maple Code To Other Programming Languages Code Generation The CodeGeneration package is a collection of commands and subpackages that enable the translation of Maple code to other programming languages Languages currently suppor ted include C Fortran 77 Java MATLAB and Visual Basic For details on Code Generation refer to the CodeGeneration help page Accessing External Products from Maple External Calling External calling allows you to use compiled C Fortran 77 or Java code in Maple F
36. available for a dimension refer to the corresponding help page for example the Units length help page for the units of length Each unit has a context The context differentiates between different definitions of the unit For example the standard and US survey miles are different units of length and the second is a unit of time and of angle You can specify the context for a unit by appending the context as an index to the unit for example mile US_ survey If you do not specify a context Maple uses the default context Units are collected into systems for example the foot pound second FPS system and in ternational system or syst me international SI Each system has a default set of units used for measurements In the FPS system the foot pound and second are used to measure the dimensions of length mass and time The unit of speed is the foot second In SI the meter kilogram and second are used to measure the dimensions of length mass and time The units of speed magnetic flux and power are the meter second weber and watt respect ively Unit Conversions To convert a value measured in a unit to the corresponding value in a different unit use the Units Calculator e From the Tools Assistants menu select Units Calculator The Units Calculator application Figure 4 6 opens 130 4 Basic Computations a Maplesort Ei Units Calculator Convert between over 500 uruts of measureme
37. but represent the most commonly used packages Examples are provided to teach you how to use the different methods of calculation available in Maple including tutors assistants commands task templates plotting and context menus The examples in this chapter assume knowledge of entering commands and mathematical symbols For information see Entering Expressions page 19 For information on basic computations including integer operations and solving equations see Basic Computa tions page 101 5 1 In This Chapter Algebra page 150 Performing algebra computa Polynomial Algebra tions Linear Algebra page 159 Performing linear al Creating Matrices and Vectors gebra computations Accessing Entries in Matrices and Vectors Linear Algebra Computations Student LinearAlgebra Package Calculus page 175 Performing calculus compu Limits tations Differentiation Series Integration Differential Equations Calculus Packages Optimization page 188 Performing optimization Point and Click Interface computations using the Optimization package Efficient Computation MPS X File Support Statistics page 193 Performing statistics compu Probability Distributions and Random Variables tations using the Statistics package Statistical Computations Plotting Teaching and Learning with Maple page 198 Table of Student and Instructor Resources Student and Instructor resources for using Maple l l Student Pack
38. e Perform computations e Visually explore concepts 5 7 Teaching and Learning with Maple 201 Fd Calculus 1 Differentiation Methods File Edit Rule Definition 4oply Rule Understood Rules Help Enter a Function Function sine Variable x Sia x o The power rule has been 7 d avi g0 x sin x applied 6 j e Show Hints mla r 3 sia x x xo Constant Identity Constant Multiple Sum Difference Product uotient Power Chain Rule Integral Rewrite Exponential Natural Logarithm al hyperbolic gt lt archyperbolic gt Figure 5 11 Student Calculus1 Differentiation Methods Tutor Tutors provide point and click interfaces to the Student package functionality To launch a tutor 1 From the Tools menu select Tutors 2 Select a subject for example Calculus Multi Variable 3 Select a tutor for example Gradients Maple inserts the Student MultivariateCalculus GradientTutor calling sequence in Worksheet mode and launches the Multivariate Calculus Gradient Tutor 202 e 5 Mathematical Problem Solving By rotating the three dimensional plot you can show that the gradient points in the direction of greatest increase of the surface see Figure 5 12 and show the direction of the gradient vector in the x y plane by rotating the plot see Figure 5 13 Multivariate Calculus Gradient a File Help Plot Window Options F xASlic Pty S241 crx ly
39. gt print assign roc a option Copyright c 1990 by Waterloo Maple Inc All rights reserved local z if lt nargs and type a name function then a args 2 1 elif nargs 1 then if type a name function anything then assign internal op a elif type a then if type hs a Vist 2 name function then if nops hs a nops rhs a then zip assign internal lhs a rhs a else error ambiguous multiple assignment end if else error invalid arguments end if elif type a set list then map procname a else error invalid arguments end if else seq procname 1 1 args end if NULL end proc Modules Maple procedures associate a sequence of commands with a single command The module a more complex programming structure allows you to associate related procedures and data A key feature of modules is that they export variables This means that the variables are available outside the module in which they are created Most Maple packages are implemen ted as modules The package commands are exports of the module 9 5 Programming in Documents 387 For more information on modules refer to the module help page 9 5 Programming in Documents To write Maple code you could simply open a Maple worksheet and start typing However if you want to create a readable document with the code interspersed or hidden
40. i a a Fa 2 0 120 04 7 l 9 l 1 x e a 0 2 P270 120 6 cocoon 318 7 Creating Mathematical Documents Insert a table with 1 row and 2 columns Enter the information in text and executable 2 D Math to create the calculation and plot as shown Table Settings In the Properties dialog Table Properties menu 1 Set Exterior and Interior Borders to None 2 Hide Maple input and execution group boundaries Clear the Show input and Show ex ecution group boundaries check boxes Using the Table menu 3 Change row Alignment to Center 7 5 Canvas Using the drawing tools you can sketch an idea in a canvas draw on plots and draw on images See Figure 7 15 For details about the drawing feature refer to the 7DrawingTools help page 7 5 Canvas 319 Text Math Piot Animation PrTT OG09 MCE WX gt HH Constant Gain Feedback Figure 7 15 Drawing Tools and Canvas Insert a Canvas To insert a canvas 1 Place the cursor where the canvas is to be inserted 2 From the Insert menu select Canvas A canvas with grid lines appears in the document at the insertion point The Drawing icon is available and associated context bar icons are displayed The tools include the following selection tool pencil free style drawing eraser text insert straight line rectangle rounded rectangle oval diamond alignment drawing outline drawing fill drawing linestyle and drawing canvas propert
41. ji 3x 1 x 0 To expand the numerator and denominator use the expanded option 2 2 xy gt normal 2 expanded x y XTY gees a 5 ARY FY gt normal sinfi x l sin x Sorting To sort the elements of an expression e Use the sort command The sort command orders a list of values or terms of a polynomial gt sort 4 3 2 1 4 43 0 4 0 2 1 3 4 43 gt sort x 4 7x 1 9x 5 gt te O 9 a Hat gt sort xy 6y x 2V ee oe 6x7 2y syty For information on sorting polynomials see Sorting Terms page 152 For more information on sorting refer to the sort help page 8 3 Working with Maple Expressions 359 Evaluating Expressions Substituting a Value for a Subexpression To evaluate an expression at a point you must substitute a value for a variable To substitute a value for a variable using context menus 1 Right click Control click for Macintosh the expression Maple displays a context menu 2 From the context menu select Evaluate at a Point The Evaluate at a Point dialog is displayed See Figure 8 2 W Evaluate at a Point Evaluate the expression at the point Figure 8 2 Evaluate at a Point 3 In the drop down list select the variable to substitute 4 In the text field enter the value to substitute for the variable Click OK In Worksheet mode Maple inserts the eval command calling sequence that performs
42. key and then press the equal sign key To the right of the expression Maple inserts an equal sign and then the value of the expres sion 2 7 _ amp 9 7 7 1 99 You can replace the inserted equal sign with text or mathematical content To replace the equal sign 1 Select the equal sign Press Delete 2 Enter the replacement text or mathematical content For example you can replace the equal sign with the text is equal to 68 2 Document Mode T 11 oj 85 is equal to D9 In mathematical content pressing Enter evaluates the expression and displays it centered on the following line The cursor moves to a new line below the output 2 d 9 TI 85 99 2 1 By default Maple labels output that is generated by pressing Enter For information on equation labels see Equation Labels page 97 In this manual labels are generally not displayed In text pressing Enter inserts a line break You can use the basic algebraic operators suchas and with most expressions in cluding polynomials see Polynomial Algebra page 150 and matrices and vectors see Matrix Arithmetic page 169 PE x4 1 9 4 9x 12 7 3x ll 4 8 99 27 69 29 81 207 87 9 F a 2 5 Editing Expressions and Updating Output One important feature of Maple is that your documents are live That is you can edit expres sions and quickly recalculate results To update one computation 1 Edi
43. r 14 10 cos 10 x dx 18 92510790 Display Volume C Disks C Both Line of Revolution Horizontal Vertical Distance of rotation line From coordinate axis OlumeotReyolutionf1l 10 cos 1lO0 x O 6 axis horizgontal distancetromaxis Maple Command 0 output plot Figure 5 15 Volume of Revolution Tutor After you Close the tutor the plot is inserted into your worksheet Check for Existing Tools Task Template 1 From the Tools menu select Tasks and then Browse The Browse Tasks dialog opens displaying a list of tasks in the left pane The tasks are sorted by subject to help you quickly find the desired task 2 Expand the Calculus Integral Applications Solids of Revolution folder 3 From the displayed list select Volume The Volume of Revolution task is displayed in the right pane of the Browse Tasks dialog 210 5 Mathematical Problem Solving 4 Select the Insert into New Worksheet check box 5 Click Insert Default Content Before inserting a task Maple checks whether the task variables have assigned values in your worksheet If any task variable is assigned the Task Variables dialog opens allowing you to modify the names Maple uses the edited variable names for all variable instances in the inserted task The content is inserted into your docu ment See Figure 5 16 Volume of Revolution Calculate the volume of revolution for a solid of revolution when a fu
44. relatively or in units of the last digit For more information on uncertainty specification refer to the ScientificErrorAnalysis Quantity help page The output displays the value and uncertainty of the quantity gt with ScientificConstants with ScientificErrorAnalysis gt Quantity 105 1 2 Quantity 105 1 2 gt Quantity 105 0 03 relative Ouantity 105 3 15 4 18 To specify the error in units of the last digit the value must be of floating point type gt Quantity 105 0 12 uld Quantity 105 0 1 2 To access the value and uncertainty of a quantity with uncertainty use the evalf and Scien tificErrorAnalysis GetError commands gt evalf 4 18 gt GetError 4 18 3 15 Rounding To round the error of a quantity with uncertainty use the ApplyRule command For a description of the predefined rounding rules refer to the ScientificErrorAnalys is rules help page 4 5 Units Scientific Constants and Uncertainty 141 gt GetError ApplyRule 4 18 round 2 os Units Quantities with errors can have units For example the scientific constants and element and isotope properties in the ScientificConstants packages are quantities with errors and units To construct a new quantity with units and an uncertainty include units in the Quantity calling sequence For an absolute error you must specify the units in both the value and error gt with Units Standard wit
45. the fraction 10 Enter the symbol The cursor moves to the denominator with the entire expression in the numerator _ Coa 1 1 Introduction to Maple 9 12 Press the right arrow key to move right and out of the denominator aa k 2 position To evaluate the expression and display the result inline 13 Press Ctrl Command Macintosh To execute 2 D Math you can use any of the following methods e Pressing Ctrl Command for Macintosh That is press and hold the Ctrl or Command key and then press the equal sign key This evaluates and displays results inline e Pressing the Enter key This evaluates and displays results on the next line and centered e Right click Control click for Macintosh the input to invoke a context menu item From the context menu select Evaluate and Display Inline See Context Menus page 39 for more details e Using the context menu item Evaluate Toolbar Options The Maple toolbar offers several buttons to assist you when interacting with Maple See Table 1 2 Table 1 2 Maple Toolbar Options Basic Usage Equivalent Menu Option or Command Inserts plain text after the current execu T From the Insert menu select Text tion group Inserts Maple Input after the current exe gt From the Insert menu select Execution cution group For details refer to Execu Group and then After Cursor tion Groups page 19 From the Format menu select I
46. 0 0 Right Margin 0 0 First Line 0 0 Below 0 0 Alignment Left w Bullets and Numbering Style None Initial List Value Bullet Suffix _ Page Break Before Linebreak Space Figure 7 3 Paragraph Style Dialog For example in the pasted text select all of the items under Parameters then open the Paragraph Style dialog Notice that the spacing has already been set In the Indent section change the Left Margin indent to 10 0 pt 288 7 Creating Mathematical Documents In the Bullets and Numbering section click the Style drop down and select Dash Click OK to close the dialog and apply the styles Result plot create a two dimensional plot Calling Sequence plott plott x x0 x1 plotivl 72 Parameters f expression in independent variable x F independent variable 0 zl left and right endpoimts of horizontal range i 72 x coordinates and y coordinates For more information refer to the paragraphmenu help page Character and Paragraph Styles Maple has predefined styles for characters and paragraphs A style is a set of formatting characteristics that you can apply to text in your document to change the appearance of that text When you apply a style you apply a group of formats in one action e A character style controls text font size color and attributes such as bold and italic To override the character style within a paragraph style you
47. 1 4 cos 4 theta Change the x axis range 2 In the Select Plot Type window a With 2 D polar plot selected change the Angle of theta to 0 8 Pi In the Plot Options window 3 From the Color group box select Magenta Plot the expression 4 Click Plot To see the Maple syntax used to generate this plot see Maple commands from Creating Plots Interactive Plot Builder page 250 Example 6 Interactive Plotting Using the Interactive Plot Builder you can plot an expression with several of its variables set to numeric values The Interactive Parameter window allows you to interactively adjust these numeric values within specified ranges to observe their effect To access this window enter an expression with two or more variables and select Interactive Plot with x parameter from the Select Plot Type and Functions drop down menu 6 2 Creating Plots 245 Interactive Parameter Maplet File Help Plot Window abl Parameters 0 1 E ye Flot Command plot sini 5 000000000 x 7 x 2Z 6 10 10 labels x 7 Cancel Command Figure 6 1 Interactive Parameter Window Launch the Interactive Plot Builder and enter an expression 1 Add the expression x 3 sin x t 246 6 Plots and Animations In the Select Plot Type window 2 From the Select Plot group box select Interactive Plot with 1 parameter 3 Change the range of the x axis to 0 5 4 Change the t range to 0 10 5
48. 108 Bohr radius 135 bold format 284 bookmarks using 327 boolean expressions 362 370 377 brackets angle 159 161 break statement 379 browser Matrix 163 342 Task 91 bullets format 286 button embedding 329 Button component 389 by clause 374 excluding 374 negative 375 C CAD Link Assistant 37 calculus 187 clickable problem solving 235 multivariate 186 Student package 187 of variations 187 packages 186 study guides 199 teaching 187 199 vector 186 Student package 187 calling sequence 83 canvas inserting 319 canvas style sketch pad 321 caret entering 110 central tendency 139 character styles creating 290 description 288 Check Box component 390 Cholesky decomposition 171 Classic Worksheet tables 315 Classic Worksheet Interface xili clickable math 235 Code Edit Region 387 CodeGeneration package description 86 coeff command 156 coefficients polynomials 156 coeffs command 157 collect command 156 colon 81 82 color of plots 267 combine command 355 errors option 142 Combo Box component 390 command completion 7 48 Command line Interface xii commands 85 and task templates 91 displaying procedures 385 entering 46 help 54 hiding 387 388 iterative 381 mapping over set or list 382 package 84 top 84 top level 82 compatibility Index 427 worksheet 335 complex expressions 362 complex numbers 30 compoly command 157 components adding
49. 135 Favorites palette 22 files image formats 322 reading from 415 writing to 413 fill option 165 finite fields 109 solving equations 126 finite rings 109 floating point computation 103 accuracy 105 hardware 105 significant digits 104 numbers 102 rational approximation 90 Flux command 187 font color 284 foot pound second FPS system 75 129 footers 297 for from loops 373 for in loops 375 formal power series solutions 125 format labels 50 Format menu bookmarks 328 quick formatting 284 frac command 146 fractions approximating 71 entering 6 frequency plot 195 Frobenius form matrix 173 from clause 374 excluding 374 fsolve command 117 full evaluation 365 367 FunctionAdvisor command 83 functional operators 343 differentiating 179 plotting 346 versus expressions 344 functions converting between 357 defining as functional operators 343 G Gaussian elimination 173 Gaussian integers 110 GaussInt package 110 gcd command 158 gcdex command 158 Global Optimization Toolbox 188 global variables 385 glossiness of 3 D plots 267 go to bookmark 329 gradient 202 Gradient Tutor 201 Graphing Calculator xiii greatest common divisor 108 158 H Handwriting palette 28 has command 349 hastype command 349 HazardRate command 195 headers 297 Help Navigator Using 56 help page adding hyperlink to 326 help system accessing 54 description 58 Edit menu
50. 2 1 In This Chapter Introduction page 63 e Comparison of Document and Worksheet Modes Entering Expressions page 64 Overview of e Palettes tools for creating complex mathematical expres Symbol Names sions Mathematical Functions Evaluating Expressions page 67 How to eval Displaying the Value Inline a a Displaying the Value on the Following Line Editing Expressions and Updating Out Updating a Single Computation put page 68 How to update expressions and Updating a Group of Computations regenerate results Updating All Computations in a Document Performing Computations page 69 Overview Computing with Palettes of tools for performing computations and solving problems Context Menus Assistants and Tutors 2 2 Introduction Maple has two modes Document mode and Worksheet mode Document mode is designed for quickly performing calculations You can enter a mathem atical expression and then evaluate manipulate solve or plot it with a few keystrokes or mouse clicks This chapter provides an overview of Document mode Document mode sample Find the value of the derivative of Inl 1 at x 4 differentiate w r t x 2x evaluate at point 8 mht M SC 63 64 2 Document Mode l over the interval 0 7 Integrate sin it sin dx sin x Ci 0 Worksheet mode is designed for interactive use through commands and programming using the Maple language The Wor
51. 2 Creating Plots 247 xy 1 Enter and evaluate an expression for example 2 Right click Control click for Macintosh the expression 3 From the context menu select Plots 3 D Plot x y x y xy gt 6 1 ery os 248 6 Plots and Animations Copy as MathML Numeric Formatting Explore ppl a Command Assign to a Mame Collect b Combine b Complete Square b Denominator Differentiate Differentiate Implicit Evaluate at a Point Expand Factor Integrate b Limit Mormal Mumerator Sequence Series Simplify FT F F F Solve Complex Maps Constructions Conversions Integer Functions Integral Transforms Language Conversions Optimization Plots Sorts Units 3 D Plot ayy 2 D Implicit Plot P Yi k 3 D Implicit Plot P Plot Builder Tr Fr F F 0 F F F F F For information on customizing plots using the context menu see Context Menu Op tions page 264 6 2 Creating Plots 249 Dragging to a Plot Region To use the drag and drop method use the plot region created by one of the other methods or insert an empty plot region into the document Empty plot regions can be two dimensional or three dimensional Advantages of the drag and drop method include the ease of adding and removing plots and the independence from plotting command syntax Example 1 From the Insert menu select Plot 2 D 2 Enter the expression sin x in an input region 3
52. 4 6 Restricting the Domain 143 4 6 Restricting the Domain By default Maple computes in the complex number system Most computations are per formed without any restrictions or assumptions on the variables Maple often returns results that are extraneous or unsimplified when computing in the field of complex numbers Using restrictions you can more easily and efficiently perform computations in a smaller domain Maple has facilities for performing computations in the real number system and for applying assumptions to variables Real Number Domain To force Maple to perform computations in the field of real numbers use the RealDomain package The RealDomain package contains a small subset of Maple commands related to basic precalculus and calculus mathematics for example arccos limit and log and the symbolic manipulation of expressions and formulae for example expand eval and solve For a complete list of commands refer to the RealDomain help page After you load the RealDomain package Maple assumes that all variables are real Com mands return simplified results appropriate to the field of real numbers gt with RealDomain gt simplif x x gt Infe Some commands that generally return NULL instead return a numeric result when you use the RealDomain package g gt 32 2 Complex return values are excluded or replaced by undefined gt solve x 1 144 4 Basic Computations gt arcsin e
53. 5 126 4 Basic Computations gt infolevel pdsolve 5 gt pdsolve 4 14 Checking arguments First set of solution methods general or quase general solution Second set of solution methods complete solutions Trying methods for first order PDEs Second set of solution methods successful f xy _FI y For more information on solving PDEs including numeric solutions and solving PDE sys tems refer to the pdsolve help page Integer Equations To find only integer solutions to an equation use the isolve command The isolve command finds solutions for all variables For more information refer to the isolve help page gt isolve x y 13 x Zl y Zi 13 Integer Equations in a Finite Field To solve an equation modulo an integer use the msolve command The msolve command finds solutions for all variables For more information refer to the msolve help page gt msolve x 1 13 Solving Linear Systems To solve a linear system use the LinearAlgebra LinearSolve command The LinearSolve command returns the vector x that satisfies A x B For more information refer to the LinearAlgebra LinearSolve help page For example construct an augmented matrix using the Matrix palette see Creating Matrices and Vectors page 159 in which the first four columns contain the entries of A and the final column contains the entries of B 4 4 Solving Equations 127 59 44 17 1 l 10 2
54. 5 17 1 15 1937487 9 4 Procedures 385 For more information on writing procedures including options and local and global variables refer to the procedure help page Procedure Return Values When you run a procedure Maple returns only the last statement result value computed Maple does not return the output for each statement in the procedure It is irrelevant whether you use semicolons or colons as statement separators proc a b b D end proc gt p a a gt p 1 2 Displaying Procedure Definitions Unlike simple Maple objects you cannot display the value of a procedure by entering its name gt geometric_mean geometric_mean You must evaluate the name of the procedure using the print or eval command gt print geometric_mean proc x y sqrt x y end proc Displaying Maple Library Procedure Definitions Maple procedure definitions are a valuable learning tool To learn how to program in Maple it is recommended that you examine the procedures available in the Maple library By default the print command returns only the proc and end proc statements and if present the description fields of a Maple procedure 386 9 Basic Programming gt print assign proc a end proc To display a Maple library procedure definition first set the value of the interface verb oseproc option to 2 Then re execute the print calling sequence gt interface verboseproc 2
55. 57 Help Navigator 55 manuals 56 search 56 table of contents 56 tasks 56 topic search 56 tutorials 56 View menu 57 Hermitian transpose matrix and vector 171 Hessenberg form 173 hexadecimal numbers 109 hide worksheet content 298 highlight color 284 Hilbert Matr x 173 histogram 196 How Dol topics 58 hyperlinks in worksheet 323 entering 30 111 Index 431 icons Open as example worksheet 57 if statement 370 ifactor command 106 108 354 igcd command 108 images adding hyperlink to 324 file format 322 inserting 322 imaginary unit entering 30 111 implied multiplication 6 implies operator 370 Import Data Assistant 36 414 indent format 286 indeterminates 352 indets command 352 indices 83 167 inequations solving 112 for real solutions 143 symbolically 114 infinite loops 379 infolevel command 125 input 1 D Math 81 2 D Math 80 prompt 80 separating 82 setting default mode 81 insert bookmark 328 hyperlink 324 images 322 section 296 sketch pad 319 table 306 Installer Builder Assistant 37 instructor resources 211 Int command 185 int command 185 integers 432 Index commands 107 computations 109 context menu 89 factoring 106 Gaussian 110 modulo m 109 solving equations 126 solving modular equations 126 integration 69 88 183 definite 184 functional operators 347 indefinite 183 iterated 185 line 185
56. Builder Select Plot Builder Select Plot Type and Functions Edit Functions Select Plot 2 D plot 2 D polar plot 3 D conformal plot of a complex valued Function 2 D conformal plot of a complex valued Function 2 D complex plot 3 D complex plot Select Variable Purposes Ranges and Plot Options 4 Modify the plot range to x 0 to 2 Pi Select Variable Purposes Ranges and Plat Options On Plot return plot command 5 Click Plot to display the plot in the document 6 From the graph we can see all of the solutions within the interval 0 27 To approximate the values click on the plot then use the Point Probe tool to view the coordinates of the cursor in the toolbar 5 8 Clickable Math Examples 225 Solution by Task Template 1 From the Format menu select Tasks 5 Algebra Browse Expand the Algebra folder and gt Complete the Square select Solve Analytically in a Specified Complex Arithmetic Interval betes Conic Analysis and Graph seee Solve a Set of Equations Symbolically seen Solve an Equation Numerically i seee Salve an Equation Symbolically Solve an Inequality Solve Analytically in Specified Interval 2 Click Insert Minimal Content Solve Analytically in a Specified Interval Find the roots ina gt Student Calculus Roots 15 0 2 2 specified interval aif Ss of 3 fi arcsin arcsin T arcsin 16 4 3 T arcsin 20 Expr
57. D Math input you can change the default math input notation To change math input notation for a session or globally across all documents 1 From the Tools menu select Options The Options Dialog opens 22 e 1 Getting Started 2 Click the Display tab 3 In the Input Display drop down list select Maple Notation 4 Click the Apply to Session or Apply Globally button Important The new input display becomes the default setting after pressing the Enter key Palettes Palettes are collections of related items that you can insert into a document by clicking or drag and dropping The Maple environment provides access to over 20 palettes containing b items such as symbols layouts 4 mathematical operations l f a and much cf more By default palettes are displayed in the left pane of the Maple environment when you launch Maple If the palettes are not displayed 1 From the View menu select Palettes 2 Select Expand Docks 3 Right click Control click Macintosh the palette dock From the context menu select Show All Palettes Alternatively from the main menu select View Palettes Arrange Palettes to display specific palettes You can create a Favorites palette of the expressions and entities you use often by right clicking Control click Macintosh the palette template you want to add and selecting Add To Favorites Palette from the context menu 1 2 Entering Expressions 23 Table 1 7 Pal
58. Darei Valles A E E i qx x z J a 2 Ble f A 9 Display Gradient Field Plot Plat Options lGradient eS ecetycetly x y 2 1 output plot Figure 5 12 Multivariate Calculus Gradient Tutor Maple Command ME Multivariate Calculus Gradient File Help Plot Window Maple Command Gradient 3 x 2 y 24 1 Figure 5 13 Multivariate Calculus Gradient Tutor Showing x y Plane x 5 7 Teaching and Learning with Maple Options Pe AS29241 x joy dete Values Atx 3 2 1 f grad Display Gradient Field Plot Plot Options 2 1 output plot When you close the tutor Maple inserts the 3 D plot 204 5 Mathematical Problem Solving gt Student MultivariateCalculus GradientTutor _ Many Student package commands can return a value mathematical expression plot or animation This allows you to compute the final answer see the general formula applied to a specific problem or visualize the underlying concepts For example the Student VectorCalculus LineInt line integral command can return the following e Plot that visually indicates the vector field path of integration and tangent vectors to the path e Unevaluated line integral e Numeric value of the line integral gt with Student VectorCalculus 5 7 Teaching and Learning with Maple 205 gt Linelnt VectorField lt y
59. Enter x 9 To evaluate the integral press Enter 3 10 Equation Labels Execution Groups with Multiple Outputs An equation label is associated with the last output within an execution group gt 2 cos z TAT 0 3265306122 zya gt 3 7 y u Label Numbering Schemes You can number equation labels in two ways e Flat Each label is a single number for example 1 2 or 3 e 99 3 7 3 8 e Sections Each label is numbered according to the section in which it occurs For example 2 1 is the first equation in the second section and 1 3 2 is the second equation in the third subsection of the first section To change the equation label numbering scheme e From the Format menu select Equation Labels Label Display In the Format Labels dialog Figure 3 6 select one of the formats e Optionally enter a prefix 100 3 Worksheet Mode gt sin x dx l cos x Cuestion 1 gt z Questioni dx Format Labels Label Numbering Prefix Question Label Wumbering Scheme Flak Mumeric Ww Figure 3 6 Format Labels Dialog Adding a Prefix Features of Equation Labels Although equation labels are not descriptive names labels offer other important features e Each label is unique whereas a name may be inadvertently assigned more than once for different purposes e Maple labels the output values sequentially If you remove or insert an output Maple automatically re
60. Evaluate and Display Inline Ctr Explore k Apply a Command Assign to a Mame Collect b Combine b Differentiate b Evaluate at a Point Integrate b Lirit Sequence Series Simplify r Fr F F Salve Complex Maps Constructions Conversions Integer Functions Integral Transforms Language Conversions Plots Units Tr Fr F0 F F0 F0 F F Help on Command 2 D Math b After ae do Plot the expression sin x and its derivative sin x cos x in the same plot dx Figure 7 10 Working with Document Blocks 302 e 7 Creating Mathematical Documents Result plot create a two dimensional plot Calling Sequence plot f x plot f x x0 x1 plotivl v2 Parameters f egpression in independent variable x F independent variable 0 zl let and night endpoints of horizontal range xl 2 x coordinates and y coordinates Plot the expression sin x and its derivative sin x cos x in the same plot Inline Document Output Document blocks can display content inline that is text input and output in one line as presented in business and education documents In document mode content 1s displayed inline by default To display content inline 1 Place the cursor in the document block 2 From the View menu select Inline Document Output View Document Code To view the contents that is all code and expanded execution groups within a document block you must expand the
61. Insert menu option that plot is not considered output Therefore if you clear the Output check box in the Show Contents dialog that plot will be visible in the document e To hide an inserted plot clear the Graphics check box in the Show Contents dialog Inserted images and the Canvas are not considered output As such they are not hidden if you clear the Output check box e To hide an inserted image or canvas clear the Graphics check box in the Show Contents dialog Indentation and the Tab Key The Tab icon allows you to set the Tab key either to move between placeholders or to indent For example with the Tab icon off click the exponent button in the Expression palette The expression 1s inserted with the first placeholder highlighted To move to the next placeholder use the Tab key Tab icon off Allows you to move between placeholders using the Tab key H a The Tab icon is disabled when using 2 D Math Math mode and as such the Tab key allows you to move between placeholders Tab icon on Allows you to indent in the document using the Tab key H 300 7 Creating Mathematical Documents 7 3 Commands in Documents Document Blocks With document blocks you can create documents that present text and math in formats similar to those found in business and education documents In a document block an input prompt or execution group is not displayed By hiding Maple input such that only text and results a
62. MapleSim model For example you 11 5 Connectivity 423 can use Maple commands and tools to manipulate your model equations develop custom components based on a mathematical model and visualize simulation results MapleSim software 1s not included with the Maple software For more information on MapleSim visit http www maplesoft com maplesim Sharing and Storing Maple Worksheet Content The MapleCloud You can use the MapleCloud to share worksheet content with other users view content shared by other users and store entire standard Maple worksheets or selected content from standard Maple worksheets Through the MapleCloud palette you can upload standard Maple worksheet content and allow other users to download a copy of that content You can also upload and store content in a user specific area that only you can access A list of shared worksheets that you have permissions to view are displayed in the Maple Cloud palette To share content with specific users you can either create a user group or select an existing group and allow only those group members to access your content For more information about groups see worksheet cloud groups Users need an internet connection to use the MapleCloud To share worksheet content create manage and join user groups and view group specific content you must log in to the MapleCloud using a Maplesoft com Gmail or Google Mail account name and password A Maplesoft com membersh
63. MathML Paste Ctrl V Explore pply a Command Differentiate b Evaluate at a Point Expand Integrate I Left hand Side Manipulate Equation Map Command Onto Move to Left Moye bo Right b Megate Relation Right hand Side Sequence I Simplify b Solve Test Relation Conversions d Integral Transforms b Flots b 2 D Math d Result move to left 5x 7 3x 42 2x 9 0 A brief description move to left is displayed above the arrow that connects the input and output 3 Right click the output from the previous action 2x 9 0 and select Solve Isolate Expres sion for x Input move to left x 7 3x4 2 2 x_ 9 0 Result ee el Pei te aa move to left 1 1 Introduction to Maple 15 Copy as MathML Numeric Formatting Explore Appl a Command Differentiate b Evaluate at a Point Expand Integrate b Left hand Side Manipulate Equation Map Command Onto Move bo Right Megate Relation Right hand Side Sequence b Simplify b Solve b Test Relation Conversions b Integral Transforms b Plots b isolate for x x 9 0 Isolate Expression For x Numerically Solve b Numerically Solve From point Obtain Solutions For Solve Solve explicit Solve general solution Solve For variable b tu o 16 1 Getting Started Now that we have solved the equation we can plot it To do this we will copy the equation 2x 9 0 to anew document bl
64. Methods Tutor page 231 e Solution by First Principles page 232 Immediate Evaluation of the Integral l 1 Enter the integral JS dy ina blank 4 x document block region 2 Right click the expression and select Evaluate and Display Inline dx _ arcs n 3 3 5 8 Clickable Math Examples 231 Solution by Integration Methods Tutor Result in Document 1 Load the Student Calculus 1 package From Loading Student Calculus1 the Tools menu select Load Package Stu dent Calculus 1 2 Ctrl drag the integrand blank document block region 3 Right click the expression and select Tutors Calculus Single Variable Integration Fie Edit Rule Definition Apply Rule Understood Rules Help Methods The Integration Methods Tutor dis Enter a function Function 1 4 2 1 2 Variable x From to plays Click on any button ta applya rule Show Hints Constant Constant Multiple Difference Parts Partial Fractions Change Exponential Natural Logarithm lt brig gt iv lt hyperbolic gt iw 4 Perform a change of variables by selecting Change and entering x 2 sin u The change rule has been applied Show Hints Partial Fractions 232 e 5 Mathematical Problem Solving 5 Apply the constant rule by clicking Constant The revert rule has been 7 pe 6 To revert back to the original variable click 4 x dx appliea Show Hints am Revert _ f
65. NE r EENE EE AN EE EEEE EE 64 Example 1 Enter a Partial Derivative oesooereesesrresssereesssreeesssseeresseeees 65 Example 2 Define a Mathematical Function cccccceeec eee eeeeeeeseeeenes 66 ili iv Contents 24 Evaluating XC SSIONS dco eecaedadaseventacoaestaoscecnsnssenaetesmesenacmaeeteseeaseeanies 67 2 5 Editing Expressions and Updating Output cc ccc eee cc eccec ene eneeeeeeeeeneeas 68 2 6 Perrone COMIN UAN ONS sesrrseirsrperis cirvis terpei eae EAEE EESE 69 Computing with Palettes sssseisisssrrirrerrniid reiben rekrei EOR EERENS RNE REUSATE ETES CENTERS 69 COn MENUS speret renne E E E E EEE E N EERE EERE 70 Aa AAC VOLS eNEAN EEE EE EEE 76 F NO Ee MOUS aeree E EE E EE AE E EES 79 Sa JO E C ap E eaa aaa E EEE EE E E A E 79 3 2 Input Promp esrererrrirrrerinrtr h Ener EERE EENE EEEE EEEIEE EE EEEN Enke 80 SUPE 0S OE araa SE E AEEA 8l ERER o o a OE A E E 8 1 TEMPO SCV AAV ONS E cease ace A cee nee netic no ececices suse T 82 O OOS EEE EEE E A ae NE E AEE 82 The Maple PAOLA YY season tices ere gescrprctenersien eet rOn n TEER E EEE EE OER TE 82 Top Level C Ol MIA Se eden aeoavencaecurnnre ieee eine ir EER e EEEE 83 Package Commands cosets secoccsuuspassuneuesoteeesen waaeeielenerascsereieesecsnecsseceneeotss 84 DAP AICS E E AEA yesnseneman scum eee EE E E E E er enuaie sees 87 De COMIC K IVICINIS gcse ateete pessoas nese ieee seceauena sens E 89 Example 1 Using Context Menus sean sieosene
66. Plots Maple offers several methods to easily plot an expression These methods include e The Interactive Plot Builder e Context menus e Dragging to a plot region e Commands Each method offers a unique set of advantages The method you use depends on the type of plot to display as well as your personal preferences Interactive Plot Builder The Interactive Plot Builder is a point and click interface to the Maple plotting function ality The interface displays plot types based on the expression you specify The available plot types include plots interactive plots animations or interactive animations Depending on the plot type you select you can create a e 2 D 3 D plot e 2 D polar plot e 2 D 3 D conformal plot of a complex valued function e 2 D 3 D complex plot e 2 D density plot e 2 D gradient vector field plot e 2 D implicit plot 6 2 Creating Plots 239 Using the Interactive Plot Builder you can 1 Specify the plotting domain before you display the graph 2 Specify the endpoints of the graph as symbolic such as Pi or sqrt 2 3 Select different kinds of graphs such as animations or interactive plots with slider control of a parameter that is customize and display a plot by selecting from the numerous plot types and applying plot options without any knowledge of plotting command syntax 4 Apply the discont true option for a discontinuous graph The output from the Interactive Plot Builder is a plo
67. Static im tained tained by gt by gt age age 1 D 1 D 1 D 2 D Math GIF or Static m Either text PDF Format MathML or LaTeX Math if Math if Math or Math or jage or shapes depending on option selected Plot GIF Postscript Not ex Not ex Notex Not ex Static im Static im file ported ported ported ported age age Animation Animated Not expor Not ex Not ex Not ex Notex Notex Static im GIF ted ported ported ported ported ported age ted ed Hidden Not expor Not expor Not ex Not ex Notex Notex Notex Notexpor content ted ported ported ported ported ported ted Manually Not suppor Not suppor Not sup Not sup Not sup Not sup RTF Main inserted t ted ported ported ported ported page tained page break break ob ject Hyperlink Links to Plain text Plain Plain Plain Plain Plain text Plain text help pages text text text text become plain text Links to documents are renamed and conver ted to HTML links sketch out Embedded GIF Not expor Not ex Not ex Notex Not ex Static im Static im ted ported ported ported ported age age put image or Spread HTML table LaTeX Not ex Notex Notex Notex RTF Static 1m sheet tables ported ported ported ported table age 11 4 Exporting to Other Formats 419 HTML LaTeX Maple Maplet Maple Plain PDF Input Applica Text Format
68. Style Title Line Times wfe vl iz default default a defaut idefaut aa caption Symbol asta wl 10 mmes ejo vB Lz Color View none i i Constrained Scaling C Custom none nae Projection orthogonal Light Model default wt orthogonal Glossiness none iw Shading default Orientation theta 45 phi 45 psi 0 Coordinate System ea Miscellaneous cartesian wt Grid Size 25 25 a Axes SS n Transparency default none Advanced Settings SS TE Fill to xy plane wero a Je 3 Plot Options window Apply plot options Once finished you can display the plot or return the command that generates the plot to the document Example 1 Display a plot of a single variable expression Maple can display two dimensional graphs and offers numerous plot options such as color title and axis styles to customize the plot 6 2 Creating Plots 241 Launch the Interactive Plot Builder 1 Make sure that the cursor is in a Maple input region 2 From the Tools menu select Assistants and then Plot Builder Notes 1 In worksheet mode Maple inserts plots interactive in the Maple document Entering this command at the Maple prompt also opens the Plot Builder 2 Interaction with the document is disabled while the Plot Builder is running Enter an expression 3 In the Specify Expressions window a Add the expression sin x x
69. The following example shows the plotting command returned by the example in Interactive Plot Builder page 270 272 6 Plots and Animations gt anima plot3d sn ese x 6 6 y 6 6 style patchnogrid lightmodel light3 shading zgrayscale scaling constrained i 1 30 i l 6 5 Creating Animations 273 Animate a 2 D plot a Tt gt animate plot 5 cos 2 0 8 0 t coords polar t 2 1 frames 50 P be f 0 78540 2 For more information on the animate command refer to the plots animate help page The plot3d viewpoint Command You can use the viewpoint command to create an animation in which the position from which you view a 3 D plot moves in all directions and in various angles around the plot surface based on coordinates and parameters you specify This type of animation creates the effect of flying through around beside towards and away from a plot surface in three dimensional space The moveable position from which you view the surface is called the camera You can specify the orientation of the camera to view different sides of a surface the path along which the camera moves throughout and around a surface and the location of the camera in 3 D space in each animation frame For example you can specify coordinates to move the camera to specific points beside a surface a pre defined camera path to move the camera in a circle around the surface and the range of view to
70. a semicolon or colon to separate multiple inputs in the same input line gt JAA tan 3 2 2 097617696 0 05847385446 If you do not specify a semicolon or colon Maple interprets it as a single input This can either give unexpected results as below or an error gt 4 4 tan 3 2 0 1226557919 3 3 Commands Maple contains a large set of commands and a powerful programming language Most Maple commands are written using the Maple programming language You can enter commands using 1 D or 2 D Math You must use 1 D Math input when programming in Maple Basic Programming page 369 provides an introduction to Maple programming To learn how to use Maple commands see the appropriate help page or use task templates For more information see The Maple Help System page 54 and Task Templates page 91 The Maple Library Maple s commands are contained in the Maple library There are two types of commands top level commands and package commands e The top level commands are the most frequently used Maple commands e Packages contain related specialized commands in areas such as calculus linear algebra vector calculus and code generation 3 3 Commands 83 For a complete list of packages and commands refer to the index help pages To access the index overview help page enter index and then press Enter For information on the Maple Help System see The Maple Help System page 54 Top Level Commands To use a t
71. apo about pletion about assumptions and properties gbout exnr e Ctrl Space Windows e Ctrl Shift Space UNIX ahs abselsal first order DF Tools abelsol ODE p Square root sqrt and then command comple tion exp and then command comple exponential function an enter exit 2 D Math F5 key 1 4 Math and Text icons in the tool bar required for products of numbers use the right arrow key to leave a denominator superscript or subscript region for more information see Command Completion page 48 For a complete list of shortcut keys refer to the 2 D Math Shortcut Keys and Hints help page To access this help page in the Maple software in Math mode enter MathShortcuts 8 1 Getting Started and then press Enter For information on the Maple Help System see The Maple Help System page 54 Example 1 Enter and Evaluate an Expression Using Keystrokes Review the following example r 2 l X Fy In this example you will enter and evaluate the expression To enter the expression 1 Enter x 2 Press Shift 6 the or caret key The cursor moves to the super script position a a 4 Press the right arrow key The cursor moves right and out of the su perscript position x y 7 Press Shift 6 to move to the superscript position 8 Enter 2 and press the right arrow key 9 With the mouse select the expression that will be the numerator of F
72. are controlled separately e You can control the visibility of interior borders by using the Group submenu of the Table menu grouping rows or columns suppresses interior borders provided that the interior border style is set by row and column group 7 4 Tables 311 For example group the columns together and group rows 2 to 4 together Then in the Table Properties dialog select Exterior Borders Top and bottom and Interior Borders By row and column group foo Z flx Plot off x and f x i 1 i 7 1 P sinf w x gioa cos co x me 5 sinf ca x a qe 2 1 1 1 sin x amp sin x cos x dx e Hidden borders are visible when the mouse hovers over a table Note You can hide the visibility of lines on mouse pointer roll over by using the View Show Hide Contents dialog and clearing the Hidden Table Borders check box Alignment Options The table alignment tools control the horizontal alignment of columns and vertical alignment of rows For column alignment the current selection 1s expanded to encompass all rows in the selected columns The alignment choice applies to all cells within the expanded selection If the document does not contain a selection the cursor position 1s used to identify the column 312 7 Creating Mathematical Documents Similarly the selection is expanded to include all columns in the selected rows for vertical alignment options The following tab
73. are rendered as static images 10 3 Creating Embedded Components 393 10 3 Creating Embedded Components Embedded Components are graphical components that you can add to your document They provide interactive access to Maple code without requiring the user to know Maple com mands and include buttons sliders math and text input areas and plot display Inserting Components The graphical interface components can be inserted by using the Components palette Figure 10 1 or by cutting copying and pasting existing components to another area of the document Although copied components have most of the same characteristics they are distinct If the Components palette 1s not visible see Palettes page 22 for instructions on viewing palettes Y Components Toggle Button Combo Box Check Box Radio Button Text Area Figure 10 1 Components Palette Editing Component Properties General Process To edit properties of components embedded in the document 1 Right click Control click for Macintosh the component to display the context menu 2 If available select Component Properties otherwise select Components Com ponent Properties The related dialog opens 3 Enter values and contents 1n the fields as necessary 394 e 10 Embedded Components and Maplets 4 For actions such as Action When Value Changes in the Slider component dialog click Edit A blank dialog opens allowing you
74. bottom right corner of the assistant window Ei Optimization Plotter Ranges Range of g hai lia extrema at 4 53559 Range of xa A extrema at 1 46441 Range of objective values default default extrema of 134 491 Plot Using Problem Domain Plot Constraints d as Surfaces Figure 5 9 Optimization Assistant Plotter Window For information on the algorithms used to solve optimization problems refer to the Op timization Methods help page 5 5 Optimization 191 Large Optimization Problems The Optimization Assistant accepts input in an algebraic form You can specify input in other forms described in the O0ptimization InputForms help page in command calling sequences The Matrix form described in the Optimization MatrixForm help page is more complex but offers greater flexibility and efficiency For example solve the linear program Maximize cx subject to Ax lt b where x is the vector of problem variables 1 Define the column vector c of the linear objective function gt with LinearAlgebra gt c RandomVector column 20 outputoptions datatype float 2 Define the matrix A the coefficient matrix for the linear inequality constraints gt A RandomMatrix 19 20 outputoptions datatype float 3 Define the column vector b the linear inequality constraints gt b RandomVector column 19 outputoptions datatype float 4 The QPSolve c
75. click Add For details see the following User Dic tionary section e To close the Spellcheck dialog and stop the spelling check click Cancel 3 When the Spellcheck is complete a dialog containing the message The spelling check is complete appears Click OK to close this dialog Note when using the Spellcheck utility you can fix spelling errors in the dialog but you cannot change the text in document The Spellcheck utility does not check grammar Selecting a Suggestion To select one of the suggestions as the correct spelling click the appropriate word from the list in the Suggestions text box If none of the suggestions are correct highlight the word in the Change To text box and enter the correct spelling Click Change to accept this new spelling User Dictionary You can create and maintain a custom dictionary that works with the Maple Spellcheck utility Properties of the Custom Dictionary File e It must be a text file that is have the file extension txt For example mydictionary txt e Itis a list of words one word per line e It is case sensitive This means that integer and Integer require individual entries in the dictionary file e It does not require manual maintenance You build your dictionary file by using the Add functionality of the Spellcheck However you can manually edit the file To specify a custom dictionary to be used with the Maple Spellcheck utility 1 Create a txt file in a directo
76. click the derivative and select Differen tiate x 214 e 5 Mathematical Problem Solving Plot the original expression xcos x 7 Insert anew document block and Ctrl drag the original expression to the new block 8 Right click the expression and select Plots gt Plot Builder 9 In the Interactive Plot Builder Select Plot Type dialog change the x Axis range to Pi to Pi Add the first and second derivatives to the plot 11 Select and then Ctrl drag the derivative of the expression onto the plot region Do the same for the second derivative 5 8 Clickable Math Examples 215 Enhance the plot by adding a legend using con a text menus cos x Curve 4 Curve 3 12 Right click in the plot region and select Le gend Show Legend 13 In the legend double click Curve 1 Notice that the Text icon is selected in the toolbar Text Delete the text and select the Math icon in the toolbar CSE This allows you to enter 2 D Math in a text region Enter the origin al expression x cos x 14 Repeat for Curve 2 and Curve 3 Add a title 15 To enter a title click the Drawing icon in the toolbar If the Drawing icon is not accessible click in the plot region 3 EE EE SESE CORS and its derivatives 16 Click T in the Drawing toolbar T 17 Click the plot region and a text region appear Notice that the toolbar has changed once again with the Text icon
77. diff command see The diff Command page 178 After defining an ODE you can solve it numerically or symbolically To solve a system numerically using the ODE Analyzer Assistant 1 Ensure that the conditions guarantee uniqueness of the solution 2 Ensure that all parameters have fixed values 3 Click the Solve Numerically button 4 In the Solve Numerically window Figure 4 4 you can specify the numeric method and relevant parameters and error tolerances to use for solving the problem 5 To compute solution values at a point click the Solve button 4 4 Solving Equations 123 Fd Solve Numerically Parameters Cutout Runge Kutta Fehlberg 4 5th order Show Function values at b 1 000000 O Cash Karp 4 5th order q 1 27526554583447 O Dverk 7 8th order q 1 30263200321421 Plot Options O Gear single step extrapolation O Rosenbrock stiff 3 4th order Livermore stiff O Boundary Value Problem solver O Taylor series lazy serie 2 Modified Extended BDF Implicit Show Maple commands Fixed step methods Soll dso0lve diftt dittiq t ti titqg c 4 cosfe t qiO 0 Digi 0 0 Numeric Absolute 1 000000e 07 default i s501111 000000 plota odeplot 3011 O 10 color redi Relative 1 000000e 06 default i On Quit Return Plot E Figure 4 4 ODE Analyzer Assistant Solve Numerically Dialog To solve a system symbolically us
78. element or isotope property For more information refer to the ScientificConstants ModifyConstant and Scienti ficConstants ModifyElement help pages You can extend the set of e Constants e Elements and isotopes e Element or isotope properties For more information refer to the ScientificConstants AddConstant ScientificCon stants AddElement and ScientificConstants AddProperty help pages For more information about constants refer to the ScientificConstants help page Uncertainty Propagation Some computations involve uncertainties or errors Using the ScientificErrorAnalysis package you can propagate the uncertainty in these values through the computation to in dicate the possible error in the final result The ScientificErrorAnalysis package does not perform interval arithmetic That is the error of an object does not represent an interval in which possible values must be contained To perform interval arithmetic use the Tolerances package For more information refer to the Tolerances help page The quantities represent unknown values with a central tendency For more information on central tendency refer to any text on error analysis for the physical sciences or engineering 140 4 Basic Computations Quantities with Uncertainty Creating To construct quantities with uncertainty use the Quantity command You must specify the value and uncertainty The uncertainty can be defined absolutely
79. element values row wise e Parameters such as shape datatype and fill that set properties of the matrix For example gt Matrix 1 2 3 4 5 6 The Matrix palette cannot fill the matrix with an arbitrary value Use the fill parameter gt Matrix 3 4 1 2 3 4 5 6 fill 2 T 9 Pa t 2 2 MH J t 3 5 6 A 2403 412 For more information on the constructors including other calling sequence syntaxes and parameters refer to the storage Matrix and Vector help pages See also Numeric Computations page 174 Accessing Entries in Matrices and Vectors Matrices To select an entry in a Matrix enter the matrix name with a sequence of two non zero integer indices row first 5 3 Linear Algebra 167 gt M 4 3 6 7 1 9 2 9 1 2 9 6 9 3 8 0 9 2 43 29 93 M 6 7 1 2 8 0 19 96 9 2 gt M 1 3 Y3 To select an entire row enter a single index to select an entire column enter first the entire range of rows 1 1 then the column index gt M 2 6 7 2 ZH gt M 1 1 1 4 3 6 7 Similarly you can access submatrices Enter the indices as a list or range gt M 2 3 1 2 ay a 19 96 Vectors To select an entry in a vector enter the vector name with a non zero integer index 168 5 Mathematical Problem Solving gt a lt 85 3 47 1 59 9 38 1 gt gt all Negative integers select entries from the end of
80. hand corner 270 6 Plots and Animations Rotate dey Rotate a three dimensional plot to see it from a different point of view gt Pan the plot by changing the view ranges for 2 D plots smartplots re sample to reflect the new view Change the position of the plot in the plot region for 3 D plots Zoom into or out of the plot by changing the view ranges for 2 D plots smartplots re sample to reflect the new view Make the plot larger or smaller in the plot window for 3 D plots Use the Selection Tool to select the information displayed in the point probe tool tooltip You can choose to display coordinates derivated from converted pixel coordinates or data points derived from the original data points 6 5 Creating Animations Animations allow you to emphasize certain graphical behavior such as the deformation of a bouncing ball more clearly than in a static plot A Maple animation is a number of plot frames displayed in sequence similar to the action of movie frames To create an animation use the Interactive Plot Builder or commands Interactive Plot Builder Creating Animations Using the Interactive Plot Builder Launch the Interactive Plot Builder and enter the expression 1 Add the expression sin i sqrt x 2 y 2 10 For information on interacting with the Interactive Plot Builder see Example 1 Display a plot of a single variable expression page 240 In the Select Plot Type window 2 From the Selec
81. keys For Windows press and hold the Ctrl key and then press the Space bar For more information on Command Completion see Command Completion page 48 Preface xv In This Manual This manual provides an introduction to the following Maple features Ease of use when entering and solving problems Point and click interaction with various interfaces to help you solve problems quickly Maple commands and standard math notation Clickable Calculus The help system Online resources Performing computations Creating plots and animations The Maple programming language Using and creating custom Maplet applications File input and output and using Maple with third party products Data structures For a complete list of manuals study guides toolboxes and other resources visit the Maplesoft web site at http www maplesoft com Audience The information in this manual is intended for first time Maple users and users looking for a little more information Conventions This manual uses the following typographical conventions bold font Maple command package name option name dialog menu or text field italics new or important concept Note additional information relevant to the section Important information that must be read and followed xvi Preface Customer Feedback Maplesoft welcomes your feedback For suggestions and comments related to this and other manuals contact doc maplesoft com 1 Getting S
82. menu In the text fields enter 0 and 1 y 0 1 Click Add Click Done to close this dialog and return to the main dialog Notice that the initial conditions are in the Conditions section 6 Click Solve Numerically A new dia log appears Piranskom i Bun sehba 49th onder D Casheiterp 4 S8h onder Ci ink eh ches heer single shep extrapolation C Rosenbrock stiff J 4th onder rieira stilt J Bendiry Fishes Probier soni Absolute 10Na i Relate 1 000000e 06 i ni Gast Gaur Pink 7 Click Solve to solve the initial value Output problem Show Function values at t Solve 0 000000 8 Click Plot to plot the solution of the DE Flot Plot Options 236 5 Mathematical Problem Solving 9 Click the Plot Options button to pi B modify the default graph if desired yi t 4y t 13 y t cos 2t solve DE interactively ss 10 Click Quit to close the ODE Analyzer and return a plot of the solution to the document 6 Plots and Animations Maple can generate many forms of plots allowing you to visualize a problem and further understand concepts e Maple accepts explicit implicit and parametric forms to display 2 D and 3 D plots and animations e Maple recognizes many coordinate systems e All plot regions in Maple are active therefore you can drag expressions to and from a plot region e Maple offers numerous plot options such as axis styles title colors shading
83. must evaluate to numeric constants For information on symbolic sums and products refer to the sum and product help pages Selecting Expression Operands The select remove and selectremove commands apply a boolean valued procedure or command to the operands of an expression For information on operands refer to the op help page e The select command returns the operands for which the procedure or command returns true e The remove command returns the operands for which the procedure or command returns false or FAIL e The selectremove command returns two expressions of the same type as the input expres sion The first consists of the operands for which the procedure or command returns true The second consists of the operands for which the procedure or command returns false or FAIL 382 9 Basic Programming The structure of the output is the same as the structure of the input See Table 9 5 For information on Maple procedures see Procedures page 383 Table 9 5 The select remove and selectremove Commands select proc_cmd expression gt select issqr 198331 889249 11751184 9857934 889249 11751184 remove proc_cmd expression gt remove var gt degree var gt 3 2 x 3 y erste 2 selectremove proc_cmd expression gt selectremove x gt evalb x gt round x sin 0 sin 1 sin 3 0 1411200081 0 0 8414709848 For information on optional arguments to
84. numerator 2 Press the forward slash key 3 Enter the denominator 4 To leave the denominator press the right arrow key To enter a power 1 Enter the base 2 Press the caret key 3 Enter the exponent which displays in math as a superscript 4 To leave the exponent press the right arrow key To enter a product 1 Enter the first factor 2 Press the asterisk key which displays in 2 D Math as a dot 3 Enter the second factor Implied Multiplication In most cases you do not need to include the multiplication operator Insert a space character between two quantities to multiply them Note In some cases you do not need to enter the multiplication operator or a space character For example Maple interprets a number followed by a variable as multiplication Important Maple interprets a sequence of letters for example xy as a single variable To specify the product of two variables you must insert a space character or multiplication 1 1 Introduction to Maple 7 operator for example x y or x y For more information refer to the 2DMathDetails help page Shortcuts for Entering Mathematical Expressions Table 1 1 Common Keystrokes for Entering Symbols and Formats Symbol Format implicit multiplication Space key x2 et F x O 4 Shift 6 or caret key exponent superscript 7 hift command symbol com Esc Macintosh Windows and a UNIX
85. of expressions in the context of finite precision arithmetic Expressions involving exact numbers for example T are replaced by close approximations using floating point numbers for example 1 41421 These computations generally involve some error Understanding and controlling this error is often of as much importance as the computed result In Maple numeric computation is normally performed if you use floating point numbers numbers containing a decimal point or the evalf command The plot command see Plots and Animations page 237 uses numeric computation while commands such as int limit 4 2 Symbolic and Numeric Computation 103 and ged see Integer Operations page 106 and Mathematical Problem Solving page 149 generally use only symbolic computation to achieve their results Exact Computations In Maple integers rational numbers mathematical constants such as z and and mathem atical structures such as matrices with these as entries are treated as exact quantities Names suchas x y my_variable and mathematical functions such as sin x and LambertW k z are symbolic objects Names can be assigned exact quantities as their values and functions can be evaluated at symbolic or exact arguments 3 gt gt T iF 1 I g 6 t3 Important Unless requested to do otherwise see the following section Maple evaluates expressions containing exact quantities to exact results
86. options surface styles and axis ranges which give you complete control to customize your plots For a reference to the types of plots available in Maple see the Plotting Guide 6 1 In This Chapter Creating Plots page 238 Interactive and command Interactive Plot Builder driven methods to display 2 D and 3 D plots Canter ien Dragging to a Plot Region The plot and plot3d Commands The plots Package Multiple Plots in the Same Plot Region Customizing Plots page 264 Methods for applying Interactive Plot Builder Options plot options before and after a plot displays Context Menu Options The plot and plot3d Command Options Analyzing Plots page 269 Plot analyzing tools Point Probe Rotate Pan Zoom Creating Animations page 270 Interactive and Interactive Plot Builder command driven methods to display animations The plots animate Command The plot3d viewpoint Command Playing Animations page 275 Tools to run anima Animation Context Bar tions 237 238 e 6 Plots and Animations Customizing Animations page 276 Methods for Interactive Plot Builder Animation Op applying plot options before and after an animation tions displays e Context Menu Options e The animate Command Options Exporting page 279 Methods for exporting plots Saving Plots to File Formats Code for Color Plates page 279 Information on Accessing Code for the Color Plates color plates 6 2 Creating
87. result and select Factor Solve for x 6 Right click on the result and select Solve Obtain Solutions for x Graphical Solution 5 8 Clickable Math Examples 221 x 7 17 Salt move to right 0 3 x 1 4 x 4 x 7 expand 0 6x 24x 18 3 right hand side O 6x 24x 18 factor bx 244 18 solutions for x 6 x 1 x 3 Now that we have seen several methods to solve this problem we can check the answer by plotting the expression 1 Ctrl drag the equation x 7 x 1 4 1 amp 4 toa new document block region and press Enter First manipulate the equation to become an ex pression 2 Right click the output and select Move to Left Note the difference in the alignment when using context menus on output rather than input The result is centered in the document with the self documenting arrow positioned at the left x 7 x 1 4 x 1 x 4 s w tu S46 17 aA w x 1 aG 4 x 4 7 move to left 222 e 5 Mathematical Problem Solving 3 Right click the output and select Left hand Side 4 Right click the output and select Expand Now that the equation is in its simplest form plot the result 5 Ctrl drag the output to a new document block 6 Right click the expression and select Plots 2 D Plot s 9 3 1 46 47 6 left hand side
88. routines to write more complicated data such as complex numbers or symbolic expressions For more information refer to the 7ExportMatrix and Ex portVector help pages For more information on matrices and vectors see Linear Algebra page 159 Saving Expressions to a File If you construct a complicated expression or procedure you can save them for future use in Maple If you save the expression or procedure in the Maple internal format Maple can retrieve it more efficiently than from a document Use the save command to write the ex pression to a m file For more information on Maple internal file formats refer to the file help page 11 2 Writing to Files 413 II 1 q gt gbinomial n k gt 4 i iig i In this example small expressions are used In practice Maple supports expressions with thousands of terms gt expr qbinomial 10 4 G qd U U 7 U expr i g i 7 U 1 4 11 3 gt nexpr normal expr nexpr q g q T g gr EET 1 d 1 q T g 7 I q J q T q gr F 1 11 4 You can save these expressions to the file qbinom m gt save gbinomial expr nexpr qbinom m Clear the memory using the restart command and retrieve the expressions using the read command gt restart gt read qbinom m gt expr i q U U U 11 5 i gU U U For more information on writing to files refer to the save help p
89. see the plot as the parameters change Exploration Assistant 1 4 Commands Even though Maple comes with many features to solve problems and manipulate results without entering any commands you may find that you prefer greater control and flexibility by using the set of commands and programming language that Maple offers The Maple Library Commands are contained in the Maple library which is divided into two groups the main library and packages e The main library contains the most frequently used Maple commands e Packages contain related commands for performing tasks from disciplines such as Student Calculus Statistics or Differential Geometry For example the Optimization package contains commands for numerically solving optimization problems For details on top level and package commands see Commands page 82 46 1 Getting Started Entering Commands If you want to interact with Maple using commands simply enter the command using 2 D math Notice that commands and variable names display in italics Maple commands are constructed in a format similar to command arguments based on the command you are using For example to factor an expression enter factor x 2x x 1 To differentiate an expression enter diff sin x x cos x To integrate an expression on the interval 0 27 enter int 2x cos x x 0 27 To plot an expression enter plot sin x x x 10 10 1
90. simultaneously impose multiple conditions on an expression specify multiple arguments in the assume calling sequence gt assume 0 lt x x lt 2 To specify additional assumptions without replacing previous assumptions use the addi tionally command The syntax of the additionally calling sequence is the same as that of the assume command gt additionally x integer about x Originally x renamed x is assumed to be 1 The only integer in the open interval 0 2 is 1 Testing Properties To test whether an expression always satisfies a condition use the is command gt assume 15 lt x 7 lt y is 100 lt xy rue The following test returns false because there are values of x and y x 0 y 10 that sat isfy the assumptions but do not satisfy the relation in the is calling sequence gt assume x nonnegint 10 lt y is l0 lt x y false To test whether an expression can satisfy a condition use the coulditbe command gt coulditbe 10 lt x y rue Removing Assumptions To remove all assumptions on a variable unassign its name gt unassign x y 146 4 Basic Computations For more information see Unassigning Names page 95 For more information on the assume command refer to the assume help page The assuming Command To perform a single evaluation under assumptions on the names in an expression use the assuming command The syntax of the assuming command 1s lt expres
91. styles 292 approximation 104 least squares 173 numeric 361 arguments 384 arithmetic 68 finite precision 102 interval 139 matrix and vector 169 modular 108 109 polynomial 150 Arrays 340 indexing 340 large 341 arrow operator 95 assign command 119 assigned command 366 425 426 Index assignment operator 94 Avogadro constant 135 Assistants Back Solver 36 CAD Link 37 Curve Fitting 34 158 Data Analysis 36 198 Equation Manipulator 36 Import Data 36 414 Installer Builder 37 Library Browser 37 Maplet Builder 37 ODE Analyzer 37 121 Optimization 37 188 overview 32 Plot Builder 37 238 Scientific Constants 37 Special Functions 37 Tools menu 33 Unit Converter 356 Units Calculator 37 129 Worksheet Migration 37 assume command 144 adding assumptions 145 and procedure variables 147 imposing multiple assumptions 145 removing assumptions 145 setting relationships between variables 144 setting variable properties 144 testing property 145 using with assuming command 146 viewing assumptions 144 assuming command 144 146 184 356 additionally option 146 and procedure variables 147 applying to all names 146 using with assume command 146 Attributes submenu character 285 paragraph 286 auto execute 304 repeating 305 security levels 306 B Back Solver Assistant 36 bar chart 195 basis 173 vector space 173 binary numbers
92. that a topic can be expanded into subtopics Question mark icon indicates a help page and displays the associated help page in the right pane when selected WS icon indicates an example worksheet Example worksheets open in a new tab in the Maple document D icon indicates a definition and displays the associated dictionary definition in the right pane when selected T icon indicates a Task template and displays the associated Task Template in the right pane when selected M icon indicates a manual Manuals open in a new tab in the Maple document Icon Jeon F 1 5 The Maple Help System 57 Viewing Help Pages as Documents In the help system examples are not executable The Maple help system allows you to open help pages as documents that you can execute To open a help page as a document or worksheet e With the help page displayed in the right pane of the help system from the View menu select Open Page as Worksheet A new worksheet tab opens and displays the help page as an executable document Viewing Examples in 2 D Math You can choose to view the examples in most help pages in either 1 D Math Maple input or 2 D Math mode The default is 1 D Math To change the math mode In the Maple help system e From the View menu select or clear the Display Examples with 2D math check box x 2 e Click the 2 D Math icon X e Note Some input in help pages displays as 1 D Math no matter which option
93. the message of the document To enter startup code for a document 1 From the Edit menu select Startup Code Alternatively click the startup code icon in the toolbar 2 Enter commands to be run each time the worksheet is opened or restart 1s called 3 Click Syntax to check the syntax of the entered code before closing 4 Click Save to save the contents and close the dialog Startup Code For Chapter09 mw File Syntax Save Syntax Figure 9 3 Startup Code Editor For more information refer to the startupcode help page Document Blocks Document blocks allow you to display the output from commands without showing the commands used You can intersperse text 2 D math and Maple commands in a readable way For more information see Document Blocks page 51 in Chapter 1 10 Embedded Components and Maplets These graphical components help you to create documents to use and share with colleagues or students that interact with Maple code within the document without needing the reader to understand that Maple code Other methods of interaction with Maple are described throughout this guide 10 1 In This Chapter Using Embedded Components page 389 Basic interact Interacting with Components ing with Maple documents containing embedded compon Printing and Exporting ents Creating Embedded Components page 393 Methods Inserting Components for creating embedded components that work together and with yo
94. the conditional expressions in order until one returns true Maple executes the corresponding statement sequence and then exits the if statement If no evaluation returns true Maple exits the if statement 372 9 Basic Programming w t 11 gt if not type x integer then printf a is not an integer x elif x gt 10 then printf Sa is an integer with more than one digit x elif x gt 0 then printf Sa is an integer with one digit x end if 11 is an integer with more than one digit Order of elif Clauses An elif clause s statement sequence is executed only if the evaluation of all previous conditional expressions returns false or FAIL and the evaluation of its conditional expression returns true This means that changing the order of elif clauses may change the behavior of the if statement In the following if statement the elif clauses are in the wrong order gt if not type x integer then printf Sa is not an integer x elif x gt 0 then printf Sa is an integer with one digit x elif x gt 10 then printf Sa is an integer with more than one digit x end if 11 is an integer with one digit elif and else Clauses In an if statement with elif and else clauses Maple evaluates the conditional expressions in order until one returns true Maple executes the corresponding statement sequence and then exits the if statement If no evaluation returns true Maple executes
95. the selection commands refer to the select help page Mapping a Command over a Set or List The map command applies a name procedure or command to each element in a set or list See Table 9 6 Table 9 6 The map Command map name_proc_cmd expression gt map f a b c Sla S b S c gt map u gt int cos x x 0 u Pi 4 Pi 7 Pi 3 0 p J2 cos T r 0 8660254038 ot For information on mapping over the operands of other expressions optional arguments to the map command and other mapping commands refer to the map help page 9 4 Procedures 383 Mapping a Binary Command over Two Lists or Vectors The zip command applies a name or binary procedure or command component wise to two lists or vectors By default the length of the returned object is that of the shorter list or vector If you specify a value as the optional fourth argument it is used as the value of the missing elements of the shorter list or vector In this case the length of the return value is that of the longer list or vector See Table 9 7 Table 9 7 The zip Command Zip proc_cmd a b gt zip f i jl k 11 zip proc_cmd a b fill Rate k JUD gt zip AiryAi 1 2 0 1 31 6 r 122 AiryAi 2 1 TE For more information on the zip command refer to the zip help page Additional Information For more information on looping commands refer to the corresponding command help page
96. the value changes Toggle Button Select or display one of two options Change the images displayed and enter to code to execute when the value changes Volume Gauge Select or display an integer or floating point value Change the display and enter code to execute when the value changes Example 1 Using Embedded Components This example demonstrates several components working together to perform a task The user inputs an expression which 1s plotted when the button 1s clicked Plot options are controlled by text areas a combo box a math expression and radio buttons 392 e 10 Embedded Components and Maplets For an interactive version of this example see the mw version of this manual In Maple from the Help menu select Manuals Resources and More Manuals User Manual Enter an expression in the variable x Then click the Plot button siti x ranges Change the color peg v scaling Constrained Unconstrained Printing and Exporting a Document with Embedded Components Printing When printing a document embedded components are rendered as they appear on screen Exporting Exporting a document with embedded components to other formats produces the following results e HTML format components are exported as gif files e RTF format components are rendered as bitmap images in the rtf document e LaTeX components are exported as eps files PDF components
97. the vector gt al 1 38 To create a Vector consisting of multiple entries specify a list or range of integers in the index For more information refer to the set and range help pages gt al 1 2 gt al2 4 47 1 59 9 38 1 Linear Algebra Computations Maple has extensive support for linear algebra You can perform many matrix and vector computations using context menus Matrix operations such as multiplication and inverses can be done with the basic matrix arithmetic operators The LinearAlgebra package provides the full range of Maple commands for linear algebra and vector space computations queries and linear system solving 5 3 Linear Algebra 169 Matrix Arithmetic The matrix and vector arithmetic operators are the standard Maple arithmetic operators up to the following two differences e The scalar multiplication operator is the asterisk which displays in 2 D Math as The noncommutative matrix and vector multiplication operator is the period e There is no division operator for matrix algebra You can construct the inverse of a matrix using the exponent 1 Table 5 5 lists the basic matrix operators 93 43 19 37 48 20 19 37 gt A C 03 6 Table 5 5 Matrix and Vector Arithmetic Operators Addition L l41 63 Subtraction E Multiplication 170 5 Mathematical Problem Solving Scalar Multiplication 1116 516 228 444 Exponentiation
98. them into your document Inserting a Task into the Document To insert a task into your document 1 Select the Insert into New Worksheet check box to insert the task into a new document 2 Click one of the insert buttons e Click the Insert Default Content button Maple inserts the default content The default content level is set using the Options dialog For instructions see the usingtasks help page e Click the Insert Minimal Content button Maple inserts only the commands and embed ded components for example a button to launch the related assistant or tutor e Click the Copy Task to Clipboard button Place the cursor where you want to insert the task and then paste the task Maple inserts the default content Use this method to quickly insert a task multiple times Note You can view the history of previously inserted tasks From the Tools menu select Tasks Previously selected task names are displayed below the Browse menu item Before inserting a task Maple checks whether the task variables have assigned values in your document If any task variable is assigned the Task Variables dialog opens to allow you to modify the names Maple uses the edited variable names for all variable instances in the inserted task By default the Task Variables dialog is displayed only if there 1s a naming conflict You can Set it to display every time you insert a task To specify that the Task Variables dialog be displayed every time you i
99. there are several options available Code Edit Region The code edit region allows you to program in one contained region in a natural way Features include the ability to press Enter for line breaking and indentation preservation Figure 9 1 shows the expanded code edit region To insert a new code edit region into your worksheet e From the Insert menu select Code Edit Region Figure 9 1 Code Edit Region To execute the code within this region right click in the region and select Execute Code You can hide the code in a code edit region by minimizing the region To minimize right click in the region and select Collapse Code Edit Region When the region is minimized an icon appears with the first line of the code written next to it It is recommended that you make the first line a comment describing the program or programs contained in the region See Figure 9 2 Figure 9 2 Collapsed Code Edit Region To re execute the code in the region while it 1s collapsed click this icon For more information refer to the CodeEditRegion help page 388 9 Basic Programming Startup Code Startup code allows you to define commands and procedures that are executed each time the document is opened and after restart is called This code is completely hidden to others reading the document For example use this region to define procedures that will be used throughout the document code but that would take up space and distract readers from
100. this calculation In Figure 1 16 the solve command was used Also notice a red prompt gt before the original expression and the solve command Entering commands outside of a document block region 1s done at this input region To insert an input region click the gt button in the toolbar menu In Figure 1 16 an equation label was used to refer to the expression For more information see Equation Labels page 49 1 4 Commands 53 To collapse a Document Block e With your cursor inside the document block select View Collapse Document Block You can use this process of expanding document blocks to begin learning Maple commands Changing the Display You can specify which parts of the input and output are displayed when the document block is collapsed For each execution group in the block you can choose to display either the input or the output e Place the cursor in the execution group e From the View menu select Toggle Input Output Display Also you can choose to display output either inline or centered on a new line e From the View menu select Inline Document Output Example 9 Creating a Document Block in Worksheet Mode In Worksheet mode you can create the content using commands and then use a document block to choose how much information to display Enter the following sentence using text and 2 D Math input and output The answer to sin x dy is cos x 1 At an input prompt click the text ico
101. tion Document Approxim LaTeX en Not ex Not ex Not ex Not ex Main style ated by vironments ported ported ported ported tained HTML style and sec attributes tions LaTeX 2e macro calls MapleNet Overview of MapleNet Using MapleNet you can deploy Maple content on the web Powered by the Maple compu tation engine MapleNet allows you to embed dynamic formulas models and diagrams as live content in web pages The MapleNet software is not included with the Maple software For more information on MapleNet visit http www maplesoft com maplenet MapleNet Documents and Maplets After you upload your Maple document to the MapleNet server it can be accessed by anyone in the world using a Web browser Even if viewers do not have a copy of Maple installed they can view documents and Maplets manipulate 3 D plots and execute code at the click of a button Custom Java Applets and JavaServer Pages Technology MapleNet provides a programming interface to the Maple math engine so commands can be executed from a Java applet or using JavaServer Pages technology Embed MapleNet into your web application and let Maple handle the math and visualization Maple T A Overview of Maple T A Maple T A is a web based automated testing system based on the Maple engine Instructors can use pre written questions or create custom question banks and then choose from these questions to form quizzes and assignments Maple T A
102. to enter Maple code that 1s executed when the event occurs For details refer to the 7DocumentTools help page Removing Graphical Interface Components You can remove an embedded component by e Using the Delete key e Using the Backspace key e Placing the cursor at the component and selecting from the document menu Edit Delete Element Integrating Components into a Document Use embedded components to display information from calculations obtain input from a reader or perform calculations at the click of a button all without your readers having an understanding of Maple commands They can be entered in any part of a Maple document including a document block or table For details on each component see its help page This simple example inserts a slider with a label that indicates the current value of the slider 1 Place the cursor in the location where the embedded component is to be inserted 2 In the Components palette click the Slider item A slider is inserted into the document 3 In the Components palette click the Label item A label is inserted next to the slider x Label 4 Right click Control click for Macintosh the label component Select Component Properties The Label Properties dialog opens See Figure 10 2 10 3 Creating Embedded Components 395 Slider Properties Name SliderLabel Mame Slider1 Caption Label Tooltip Value at Lowest Position Tooltip Value at Highest Position Image
103. training materials From complete training videos to recorded training seminars to downloadable documentation you have many options to get familiar with Maplesoft products In addition whether you are an expert or someone who is considering a new license purchase a custom training session that is right for you and or your organization can be created 1 6 Available Resources 6l http www maplesoft com support training MaplePrimes A web community dedicated to sharing experiences techniques and opinions about Maple and related products as well as general interest topics in math and computing http www mapleprimes com Online Help All of Maple s help pages are available online http www maplesoft com support help Technical Support A Maple web site containing FAQs downloads and service packs links to discussion groups and a form for requesting technical support http www maplesoft com support For a complete list of resources refer to the 7MapleResources help page 62 1 Getting Started 2 Document Mode Using the Maple software you can create powerful interactive documents You can visualize and animate problems in two and three dimensions You can solve complex problems with simple point and click interfaces or easy to modify interactive documents You can also devise custom solutions using the Maple programming language While you work you can document your process providing text descriptions
104. where you can select the parameters to explore and the range for each parameter If you enter integer ranges only integer values are allowed for parameters To allow floating point values enter floating point ranges Select skip for any of the parameters to leave that parameter as a variable 4 Click Explore to continue to the Exploration Assistant The assistant opens in a new document You can use the slider or sliders to vary the parameters and see your changes as the expression output is updated 5 Once you are finished interacting with the assistant you can copy and paste the results into your document or save the interactive document for later use 44 1 Getting Started Example 8 Use the Exploration Assistant to Explore a Plot sin ax bcos x In this example we will explore how the plot of changes as we vary the parameters a and b rn me 2 Right click Control click for Macintosh the expression and se lect Explore Copy Full preci Copy as MathML Faste Ctrl Evaluate Evaluate and Display Inline Ctrl Explore NS pply a Command ssign to a Mame Collect 3 In the Explore parameter selec Expl tion dialog set the ranges a ER 0 10 0 and b 5 0 5 0 Select E floating point computation 15 0 Floating point computation 1 4 Commands 45 4 Click Explore The Explora Zz tion Assistant opens in a new imams Maplesoft document Move the sliders to
105. which to plot it document f x interval 4 Enter the new expression x cos x in the f x Enter the function f x to be evaluated and the interval on which to plot it cos x region 5 Enter the interval N N To insert the sym Interval 7 7 7 bol for pi you can use command completion or select from the Common Symbols palette 218 5 Mathematical Problem Solving 6 Click Launch Differentiation Tutor to launch Derivatives of Various Orders of the same tutor as in the previous solution fivizx cost on the Intervall Fi Pi 7 When complete click Close A plot of the ex pression and its derivatives displays in the plot 3 region of the inserted task template i on TANC fizi Ist derivative 2nd derivative Example 2 Solve for x in a Quadratic Equation Solve for x in the equation x he x je 4 x j x 4 We solve this problem using the following methods Solution through Equation Manipulator page 218 Instant Solution page 220 Step by step Interactive Solution page 220 e Graphical Solution page 221 Solution through Equation Manipulator Maple provides a dialog that allows you to single step through the process of manipulating an expression This manipulator is available from the context menu 1 Enter the equation x 7 x 1 4 x 1 x 4 7 ina new document block region 5 8 Clickable Ma
106. x 256y x LAN F 1024y 5 gt 1024y No Expressions You can specify expressions instead of equations The solve command auto matically equates them to zero gt solve e amp z LambertW 1 Multiple Equations To solve multiple equations or inequations specify them as a Creating and Using Data Structures page 337 gt solve Ixy y 5 x gt 0l 9 3 gt l gt gt a 5 y 5_y 5 y 5 y 5_y Yy Vy y gt solve xy y 5 x lt 0 y 5 y 5 y 5 o J a T i 7 ya 3 y y y Solving for Specific Unknowns By default the solve command returns solutions for all unknowns You can specify the unknowns for which to solve gt solve q yep A 5 a 1 l yit4es 207 I 1 yi 4es 207 2 r Ta A To solve for multiple unknowns specify them as a list 116 4 Basic Computations q r q _ l gt solve 4 5 rs solve iz y Vs ba rl s Ss 14 5s ns ee ees Sel As 4s 3 Transcendental Equations In general the solve command returns one solution to tran scendental equations gt equation sin x cos x gt solve equationl To produce all solutions use the allsolutions option gt solve equation allsolutions true t n_ Zi TEFA Maple uses variables of the form _ZN where N is a positive integer to represent arbitrary integers The tilde indicates that it is a quantity with an assumption For infor
107. x P 7 x ts of aa oe ee solutions 1 a T ET T p Using the nops command count the number of solutions gt nops solutions For more information on the nops command and operands refer to the nops help page Indeterminates To find the indeterminates of an expression e Use the indets command The indets command returns the indeterminates as a set Because the expression 1s expected to be rational functions such as sin x f x and sqrt x are considered to be indeterminate 8 3 Working with Maple Expressions 353 gt indets 3 7 x sin y Ly x y l y sin y l y To return all subexpressions of a particular type specify the type as the second argument For information on types see Testing the Type of an Expression page 348 gt indets 3 2 xX sin J 1 y radical J l y To test whether an expressions has subexpressions of a specific type without returning them use the has command For more information see Testing for a Subexpres sion page 349 Manipulating Expressions This section introduces the most commonly used manipulation commands For additional manipulation commands see Iterative Commands page 380 Simplifying To simplify an expression e Use the simplify command The simplify command applies simplification rules to an expression Maple has simplification rules for various types of expressions and forms including trigonometric functions radicals logarithmic func
108. 1 Display a plot of a single variable expression sin x gt plor x 10 10 6 2 Creating Plots 251 Example 2 Display a plot of multiple expressions in 1 variable To display multiple expressions in a plot include the expressions in a list To enter L sin x and sinc dx use the Expression palette For more information see Palettes page 22 sin x sin sin x dx a 3 ai gt plot 252 6 Plots and Animations Example 3 Display a plot of a multi variable expression sin xy _ J 3 5 5 y 5 5 view 0 0 5 lightmodel light shading X ty gt plot3d f zerayscale style patchnogrid grid 40 40 6 2 Creating Plots 253 Example 4 Display a conformal plot A collection of specialized plotting routines is available in the plots package For access to a single command in a package use the long form of the command gt plots conformal 7 Z 0 2 2 I axes normal grid 20 20 254 6 Plots and Animations Example 5 Display a plot in polar coordinates gt plots polarplot 1 4cos 4 6 0 0 8 7 color magenta 3k OK i y A Z a 6 2 Creating Plots 255 Example 6 Interactive Plotting gt plots animate plot x 3 sin x t x 0 5 0 10 f For more information on the plot options used in this section refer to the plot options and plot3d options help pa
109. 11 14016 5 0 Animation OD NOUS gxxcpeaoasdyceraetacesecn neanateadumetssoaneie ee pease tdanaeetenenseeens 275 Table 9 1 Default Clause Values cs icccicescisnadentecncentnehsavaeeneeerccsanetenainaestesaaeeianteerys 374 Table 9 2 erative Commands conan oveenencrsanaiaenenennsnwanseensacensasenveensarsenmaeeaens 380 Table 9 3 The seq Command sssssrrireseiroreserestirenei denses Pinin r OEN EEEN EE N EEKi 380 Table 9 4 The add and mul Commands secccasicoscswscassaigiacentacawswenentnsmucaweceddatngase ses 381 Table 9 5 The select remove and selectremove Commands ccc esse eeeee eee 382 Table 9 6 The map Command sererrisrrersicrrr eritir ie scart ETON EESE NEEE EEEE EERTE 382 Table 937 Wine Ai ComimanG ssrrrerrirrean enprene e EEEE E ENE ETENEE 383 Table 10 1 Embedded Component Descriptions soosrsssosressssnnessserrrssssrersssen 389 Table 11 1 Summary of Content Translation When Exporting to Different Formats 418 Xi xii List of Tables Preface Maple Software Maple software is a powerful system that you can use to solve mathematical problems from simple to complex You can also create professional quality documents presentations and custom interactive computational tools in the Maple environment You can access the power of the Maple computational engine through a variety of interfaces Standard default A full featured graphical user interface that helps you create electronic documents to
110. 11 Input Output and Interacting with Other Products gt interface echo 2 read filename gt St n gt sum binomial n beta 2 beta 2 beta beta beta beta l n 28 n gt gt binomial 7 B BIB B 2P gt Bal 1024937361666644598071 1143287693 17982974 11 7 For more information refer to the read and interface help pages 11 4 Exporting to Other Formats Exporting Documents You can save your documents by selecting Save or Save As from the File menu By selecting Export As from the File menu you can also export a document in the following formats HTML LaTeX Maple input Maplet application Maple text plain text PDF and Rich Text Format This allows you to access your work outside Maple HTML The html file that Maple generates can be loaded into any HTML browser Exported mathematical content can be displayed in one of the following formats GIF MathML 2 0 Presentation MathML 2 0 Content or Maple Viewer and is saved in a separate folder MathML is the Internet standard sanctioned by the World Wide Web Consortium W3C for the communication of structured mathematical formulae between applications For more information about MathML refer to the 7MathML help page Maple documents that are exported to HTML translate into multiple documents when using frames If the frames feature is not selected Maple creates only one page that contains the document contents
111. 186 234 gt M E 4 S65 865 83 4 2595 865 For more information on these data structures including how to access entries and perform linear algebra computations see Linear Algebra page 159 Functional Operators A functional operator is a mapping fx gt y x The value of f x is the result of evaluating y x Using functional operators you can define mathematical functions Defining a Function To define a function of one or two variables 344 e e 8 amp 8 Maple Expressions 1 In the Expression palette click one of the function definition items See Figure 8 1 Maple inserts the function definition 2 Replace the placeholders using Tab to move to the next placeholder Note If pressing the Tab key indents the text click the Tab icon 1n the toolbar This allows you to move between placeholders 3 Press Enter ial ee a eh Figure 8 1 Function Definition Palette Items For example define a function that adds 1 to its input gt add x gt x 1 Note To insert the right arrow you can enter the characters gt In 2 D Math Maple replaces gt with the right arrow symbol In 1 D Math the characters are not replaced You can evaluate the function add1 with symbolic or numeric arguments gt add1I 12 addI x y ee te Distinction between Functional Operators and Other Expressions The expression x 1 is different from the functional operator x gt x 1 Assign the func
112. 332 dictionary 334 sqrfree command 158 Standard Document Interface xiii starting 3 Standard Units environment 132 Standard Worksheet Interface xili Startup Code 388 startup code 10 statements multiple lines 384 Statistics package 197 continuous distributions 193 description 86 discrete distributions 193 plots 195 strings 347 StringTools package 347 Student package description 87 Student Help Center 60 Student package 181 198 199 calculus subpackages 187 LinearAlgebra subpackage 174 Maplets 198 Tutors 198 student resources 211 students portal for 198 study guides 199 style set management 294 subscripts entering 7 format 284 substitute 359 sum command 381 Superscript format 284 Sylvester matrix 173 symbol completion 7 symbolic computation 102 objects 103 symbols entering 29 names 29 system of units 129 controlling 133 systeme international SI 75 129 T Tab icon 88 inserting 88 key 88 Tab icon 10 table of contents help system 56 tables 342 alignment 311 and Classic worksheet 315 appearance 310 borders 310 contents 306 execution order 315 physical dimensions 310 printing 315 using 306 visibility of cell content 314 Index 439 Task Browser 91 task template 41 task templates 91 106 128 159 175 taylor command 182 Taylor series 182 tcoeff command 156 Teacher Resource Center 60 teachers portal for 198 teaching with Ma
113. 335 Linking to a Worksheet To link to a Maple worksheet or document 1 In the Type drop down list select Worksheet 2 In the Target field enter the path and filename of the document or click Browse to locate the file Optional In the Bookmark drop down list enter or select a bookmark Note To link within a single Maple document leave the Target field blank and choose the bookmark from the Bookmark drop down list Note When linking to a custom document the path is absolute When sharing documents that contain hyperlinks ensure that target documents are in the same directory 3 Click OK 326 7 Creating Mathematical Documents Linking to a Help Page To link to a help page 1 In the Type drop down list select Help Topic 2 In the Target field enter the topic of the help page Optional In the Bookmark drop down list enter or select a bookmark 3 Click OK Linking to a Task To link to a task 1 In the Type drop down list select Task 2 In the Target field enter the topic name of the task template see the status bar at the bottom of the Task Browser window 3 Click OK Linking to a Dictionary Topic To link to a Dictionary topic 1 In the Type drop down list select Dictionary Topic 2 In the Target field enter a topic name Dictionary topics begin with the prefix Definition for example Definition dimension 3 Click OK Linking to a Maplet Application To link to a Maplet ap
114. 357 Zip command 383 Index 441 442 Index
115. 4 Commands 47 60 40 40 60 For a list of the top commands in Maple see Top Commands page 84 Package Commands There are two ways to access commands within a package using the long form of the package command or the short form Long Form of Accessing Package Commands The long form specifies both the package and command names using the syntax package com mand arguments LinearAlgebra Random Matrix 2 44 3 92 67 Short Form of Accessing Package Commands The short form makes all of the commands in the package available using the with command with package If you are using a number of commands in a package loading the entire package is recommended When you execute the with command a list of all commands in the package displays To suppress the display of all command names end the with package 48 e 1 Getting Started command with a colon Alternatively you can load packages through the Tools menu by selecting Load Package and then the package name with Optimization ImportMPS Interactive LPSolve LSSolve Maximize Minimize NLPSolve OPSolve After loading a package you can use the short form names that is the command names without the package name ESSolve x 2 x 6 x 9 12 3333333333333322 x 5 66666666666666696 For a list of the top packages in Maple see Top Packages page 86 Command Completion To help with syntax and reduce the amoun
116. 5 2 100 2 533 6l l 9 100 50 gt linearsystem og 21 3 4 2178 10 10 25 5 786 23 9 12 10 45 gt LinearAlgebra LinearSolve linearsystem 41858667400 8371733480 1674346696 209293337 For more information on using Maple to solve linear algebra problems see Linear Al gebra page 159 Solving Recurrence Relations To solve a recurrence relation use the rsolve command The rsolve command finds the general term of the function For more information refer to the rsolve help page gt rsolve fin fln 1 fln 2 A 1 AD BAY fron shve 4 bye h heed 4 10 128 4 Basic Computations 4 5 Units Scientific Constants and Uncertainty In addition to manipulating exact symbolic and numeric quantities Maple can perform computations with units and uncertainties Maple supports hundreds of units for example miles coulombs and bars and provides facilities for adding custom units Maple has a library of hundreds of scientific constants with units including element and isotope properties To support computations with uncertainties Maple propagates errors through computations Units The Units package in Maple provides a library of units and facilities for using units in computations It is fully extensible so that you can add units and unit systems as required Note Some unit operations are available as task templates see Tools Tasks Browse and through context menus Overview o
117. 86 unloading 85 page break 286 page headers and footers 297 palettes 65 69 87 359 categories 23 Components 393 favorites 22 managing 25 Matrix 159 164 overview 22 symbol recognition 28 Units 75 131 paragraph styles creating 293 description 288 parameters 384 parametric solutions 117 partial derivative entering 65 partial differential equations solving 125 paste 283 examples 57 PDEs 125 pdsolve command 125 pencil sketch pad 320 Physics package description 86 pie chart 196 piecewise command 194 Planck constant 135 Plot Builder description 37 plot command 183 Plot component 390 plot3d command 346 plots analyzing 269 pan 269 point probe 269 rotate 269 scale 269 Index 435 code for color plates 279 creating 263 context menu 246 displaying multiple plots 262 insert plot 249 Interactive Plot Builder 238 plot command 249 plot3d command 249 plots package 257 creating animations animate command 271 Interactive Plot Builder 270 plot3d viewpoint command 273 customizing 267 context menu 264 Interactive Plot Builder 264 plot options 267 plot3d options 267 customizing animations 277 command line options 277 context menu 277 Interactive Plot Builder 276 exporting 279 functional operators 346 gradient 203 line integral 204 ODEs numeric solution 123 symbolic solution 124 optimization problem 190 playing animations 275 plots package anima
118. 9 OCS Tor Color PIAS serriporierirrit arinn r n EE ORTE EEEE EEEE E 279 7 Creating Mathematical Documents ciiscascavxsonsvessesneesdeaciengencieresnedevwtansontiraneeraeed 281 Tel Ta THs CADET s oeeerriorieer ipres israp a TEESE EEEn ENEE EAEEREN EEE ET r rS Ter 281 1 2 Doc me nt Formatting marrura tees aieancn ghee venwecerraianenusobsesianeneeadsacae caucus 283 fA PAS 6G ee sree cesses ceases nee eon E eee ens ace eres 283 Quick C haracier Or Ma I o ase secession cecanaennnneanctceoirsaoyseceoutencdancatecoese ace 284 Quick Paragraph OL 1 es pt core rscravastawaseeinaen an teem ii Ten dees ere oouaea aaa 286 Character and Paragraph Styles dscccnowd encneserdnosiiornentscesinntaionopeeaduinoeaesatecedeeties 288 BS CHONG EEEIEE AE E T E E T OE E E E eres 295 Headers and Footers wcccsvierseestecnterausmncdersnsnssdtucientsenserosnncieeiworerieestnerdeseese 297 Show or Hide Worksheet Content a cevsosts cscawensuhontacuddeenisenateasessrngteaeecancaneues 298 Indentation and the Tab KEY ssssscissrsesirsisesirresisinrisisarussrdraureiirtssiiias neite esat 299 7 3 Commands in Documents een en ne ete ate ee oe eee en ee 300 Bo MeO BOCK S errereen cess eee cases a cateca EAEE EE EEEE 300 TPE E ee nee eee eee oe een rene ene E eens N 303 Contents vii UO E U e E E NEIE E A E E E E 304 TDO a a E E E I EEN E EE E EA EEEE 306 Crean a be eee ne ae EEEE E EE E nee E ene AA 306 OS OMG E E E E A T E A 306 Navig ting Tabl Cells serssrseressisersrrirer
119. AEE E E ENNE ENE O ee eee 400 Mope DOC MNE eere err eeen E E E E E EE E E E 401 IOS UOT RO WIGS sresterrris rinis eer Erei EEEREN 402 SPee MIPE EEE EEEE AE EEEE 402 Contents 1x WADI Builder eerisereerinerr tr rererere N EENT TASEN EEN EENS EES e an Edn EES 403 Maple Pa RITE eaea T EE E E ANS 407 By eV I ES E E EE E ETE RE AT E E EA TT 409 11 Input Output and Interacting with Other Products ccc cece ccc ec ene ee eee ee ees 411 IEI Ta TOC DaO eneas EEEE EEE EEE 411 e N OT ise E E EE E E EE 411 SV TAC oa T r E A tin A E ET A EEN 411 Savine Expressions to a FilE aoe asesccpssccse corectinnasacut dansaucseseceecsancsasaecteaacnas 412 LES Reading from Files ssissrrieseissesderseteserodtodernera srir Oresi EErEE EEPE EEPE E TENE 414 R adine Data from 3 Pile ieee race nace eases tienne ENE NEEE AAEE EE 414 Reading Expressions from a File ecierisiecirsserenrecrerre ariere irri 415 11 4 Exporting to Other Formats fcc nace cranes ens eorseesacaseusacesqesecbacecan i cemeesseanicens 416 FX Or ne IDO CUIMICINS seca dencms ta cnesesdecpecedastaccdenessunisaesuceevebre toners temverds 416 Maple Nei case depo er EEE EEEE EE EEE E E E E S 419 NODE TA eena E E EAEE NE 419 AO E S EEE EE E A EE I ET EE EN 420 Translating Maple Code To Other Programming Languages 085 420 Accessing External Products from Maple sicccasscccscvisdundssocanteansdvanvccencssenenet 420 Accessing Maple from External Products snsices 2casn
120. Commands Table 3 1 Top Commands or definite sum Set variable properties and relationships between variables Similar function ality is provided by the assuming command type Type checking command In many contexts it is not necessary to know the exact value of an expression it suffices to know that an expression belongs to a broad class or group of expressions that share some common properties These classes or groups are known as types series Generalized series expansion map s Apply a procedure to each operand of an expression Package Commands To use a package command the calling sequence must include the package name and the command name enclosed in square brackets package command arguments If you are frequently using the commands in a package load the package To load a package e Use the with command specifying the package as an argument 3 3 Commands 85 The with command displays a list of the package commands loaded unless you suppress the output by entering a colon at the end of the calling sequence After loading a package you can use the short form names of its commands That is you can enter the commands without specifying the package name For example use the NLPSolve command from the Optimization package to find a local minimum of an expression and the value of the independent variable at which the minimum occurs gt Optimization NLPSolve sn y l 15
121. Fil Help Plot Window Options FS xoz Point ine x Direction Bi fm of Frames 10 Values Actual Value 2 6533 Maple Command Directionalberivativel x Z y 2 x 2 output plot Figure 5 7 Directional Derivative Tutor To compute a symbolic value for the directional derivative use the Student Multivariate Calculus DirectionalDerivative command The first list of numbers specifies the point at which to compute the derivative The second list of numbers specifies the direction in which to compute the derivative For example at the point 1 2 the gradient of r points in the direction 2 4 which is the direction of greatest increase The directional derivative in the ortho gonal direction 2 1 is zero gt with Student MultivariateCalculus 182 e 5 Mathematical Problem Solving gt DirectionalDerivative x 1 x v 1 2 1 2 25 gt DirectionalDerivative x y x v 1 2 2 1 0 Series To generate the Taylor series expansion of a function about a point use the taylor command gt taylor sin 4x cos x x 0 38 3 421 5 f 3 30 x O x 4x Note If a Taylor series does not exist use the series command to find a general series ex pansion For example the cosine integral function does not have a taylor series expansion about 0 For more information refer to the Ci help page gt taylor Ci
122. GUI elements 329 palette 329 computations assistants 91 commands 85 context menus 89 errors 105 avoiding 105 integers 109 interrupting 379 linear algebra 168 mathematics 149 numeric 105 palettes 87 performing 101 149 Real number system 143 symbolic 105 syntax free 77 task templates 91 tutors 91 under assumptions 144 single evaluation 146 updating 68 with uncertainty 141 with units 132 conditional execution 370 constants 65 content command 157 context of unit 129 context menus 70 89 171 customizing animations 277 equation 112 integer 89 106 overview 39 tutors 76 using 39 convert command 356 428 Index base option 109 378 degrees option 356 mathematical functions 357 polynom option 183 set option 356 temperature option 131 units option 130 356 copy 283 examples 57 copy expressions 12 correlation 141 coulditbe command 145 covariance 141 cross product 171 Curl command 187 Curve Fitting package PolynomialInterpolation command 158 Curve Fitting Assistant 34 158 cut and paste in tables 308 D D operator 179 Data Analysis Assistant 36 198 data structures 65 337 creating 347 Database Integration 420 datatype option 165 degree command 157 polynomials 156 denom command 351 derivatives 177 directional 180 partial 65 177 prime notation 303 Tutor 199 Dial component 390 dictionary 58 198 diction
123. Index Meter component 390 min command 108 minimize 188 minimum 108 mod command 108 mod operator 109 modes Document 63 Worksheet 63 modify table 307 modp command 109 mods command 109 modular arithmetic 108 109 modules 386 MPS X files 192 msolve command 126 mul command 381 multiplication implied 6 N names 65 95 adding assumptions 144 and symbols 29 assigned 366 assigning values to 94 logical 370 previously assigned 366 protected 95 removing assumptions 145 reserved 95 unassigning 95 145 368 valid 96 versus equation labels 100 with assumptions 144 nops command 352 norm command 158 172 normal command 357 normal form 357 not operator 370 numbers 65 exact 102 floating point 102 non base 10 108 numer command 351 numeric approximation 361 computation 102 numtheory divisors command 108 O ODE Analyzer Assistant 37 121 online help 61 operands 352 selecting 381 operators 65 functional 343 logical 370 relational 370 Optimization package description 86 optimization 191 efficiency 191 plotting 190 point and click interface 188 Optimization Assistant 33 37 188 Plotter 190 Options dialog 21 or operator 370 Order environment variable 182 ordinary differential equations plotting solution 124 solving 121 orthogonal matrix 173 output suppressing 81 P packages 82 accessing commands 47 definition 45 help 54 loading 84 top
124. Maple Expressions gt eval x 5 gt eval x 1 gt eval x 2 gt eval x 3 5 For more details on levels of evaluation refer to the lastnameevaluation assigned and evaln help pages Delaying Evaluation To prevent Maple from immediately evaluating an expression e Enclose the expression in right single quotes Because right single quotes delay evaluation they are referred to as unevaluation quotes gt 4 Using an Assigned Name as a Variable or Keyword If you use an assigned name as a variable Maple evaluates the name to its value and passes the value to the command In this example that causes Maple to return an error message 8 3 Working with Maple Expressions 367 n gt 2 i Error in sum summation variable previously assigned second argument evaluates to 4 1 n Note In general it is recommended that you unassign a name to use it as a variable See Unassigning a Name Using Unevaluation Quotes page 368 To use an assigned name as a variable e Enclose the name in unevaluation quotes Maple passes the name to the command n 1 3 Important It is recommended that you enclose keywords in unevaluation quotes For example if you enclose the keyword left in unevaluation quotes Maple uses the name not its assigned value gt left 3 gt limit x 0 teft Full Evaluation of an Expression in Quotes Full evaluation of a quoted expression removes on
125. Maple User Manual Copyright Maplesoft a division of Waterloo Maple Inc 1996 2010 Maple User Manual Copyright Maplesoft Maple MapleSim Maple Application Center Maple Student Center Maplet Maple T A and Maple Net are all trademarks of Waterloo Maple Inc Maplesoft a division of Waterloo Maple Inc 1996 2010 All rights reserved No part of this book may be re produced stored in a retrieval system or transcribed in any form or by any means electronic mechanical photocopying recording or otherwise Information in this document is subject to change without notice and does not represent a commitment on the part of the vendor The software described in this document is furnished under a license agreement and may be used or copied only in accordance with the agreement It is against the law to copy the software on any medium except as specifically allowed in the agreement Adobe and Acrobat are either registered trademarks or trademarks of Adobe Systems Incorporated in the United States and or other countries Java and all Java based marks are trademarks or registered trademarks of Sun Microsystems Inc in the United States and other countries Maplesoft is independent of Sun Microsystems Inc MATLAB 1s a registered trademark of The MathWorks Inc Microsoft and Windows are registered trademarks of Microsoft Corporation NAG 1s a registered trademark of The Numerical Algorithms Group Ltd All other trademarks are t
126. Note The MmaTranslator package does not convert Mathematica programs There is a Maplet interface to the MmaTranslator package For more information refer to the 7 MmaToMaple help page Matlab Package The Matlab package enables you to translate MATLAB code to Maple as well as call se lected MATLAB functions from a Maple session provided you have MATLAB installed on your system For more information refer to the Matlab help page Accessing Maple from External Products Microsoft Excel Add In Maple is available as an add in to Microsoft Excel This add in is supported for Excel 2007 and Excel 2003 for Windows and provides the following features e Access to Maple commands from Excel e Ability to copy and paste between Maple and Excel e Access to a subset of the Maple help pages e Maple Function Wizard to step you through the creation of a Maple function call To enable the Maple Excel Add in in Excel 2007 1 Click the Microsoft Office Button and select Excel Options 2 Click Add ins 3 In the Manage box select Excel Add ins and then Go 4 Navigate to the Excel subdirectory of your Maple installation select WMIMPLEX xla and click OK 422 11 Input Output and Interacting with Other Products 5 Select the Maple Excel Add in check box 6 Click OK To enable the Maple Excel Add in for versions of Excel prior to Excel 2007 1 From the Tools menu in Excel choose Add Ins If the Maple Excel Add in is not l
127. Output 8 To display input instead of output for the first ex pression place the cursor in the first expression From the View menu select Toggle Input Output Display Only the first region displays input The answer to sin x dx is cos x 1 5 The Maple Help System The Maple program provides a custom help system consisting of almost 5000 reference pages The help system is a convenient resource for determining the syntax of Maple com mands and for learning about Maple features Accessing the Help System There are several ways to access the Maple help system e From the Help menu select Maple Help e Click B in the toolbar To get help on a specific word e In a document place the insertion point in a word for which you want to obtain help From the Help menu select Help on Alternatively press F2 Control for Macintosh to access context sensitive help 1 5 The Maple Help System 55 e In a document execute the command topic for example enter LinearAlgebra and press Enter The Maple help system opens in a separate window with two panes The left pane contains the Help Navigator where you initiate searches and browse the table of contents and the n eter Ff hre O Yoox CO Text 7 SS olye selve one or more equations edeg Moating polal arithmetic Calling Sequence felve equations variables complies Y Parameters equations PS M Apanan Re aaa aparani proc
128. Region List of Expressions To display multiple expressions in the same plot region enter the expressions in a list data structure To distinguish the surfaces apply different shading options styles or colors to each surface gt plot3d cos 5 x cos 5 y x7 3 y7 4 x 2 2 y 1 1 shading zgrayscale none color default grey style patchnogrid patch lightmodel light3 transparency 0 1 The display Command To display different types of plots in the same plot region use the display command in the plots package This example plots a curve over a hill with the shadow of the curve projected onto the hill gt z 10 G y d el gt hill plot3d z x 2 2 y 2 5 2 5 shading zhue style patchnogrid lightmodel light3 orientation 125 60 6 2 Creating Plots 263 gt xt i cos f gt yt 2sin f Maple can draw curves in three dimensional space gt curve spacecurve xt vt 10 t 0 10 color red thickness 2 gt zt i subs x xt y yt z gt shadow spacecurve xt yt zt t n m color black thickness 2 gt display hill curve shadow 264 6 Plots and Animations 6 3 Customizing Plots Maple provides many plot options to display the most aesthetically pleasing illustrative results Plot options include line styles colors shadings axis styles and titles where applic able Plot options are applied using t
129. Teaching and Learning with Maple page 198 5 4 Calculus The Task Browser Tools Tasks Browse contains numerous calculus task templates For a list of tasks navigate to one of the related folders such as Calculus Differential Equations Multivariate Calculus or Vector Calculus This section describes the key Maple calculus commands many of which are used in task templates or available in the context menus For a complete list of calculus commands refer to the Mathematics including Calculus Differential Equations Power Series and Vector Calculus subfolders and Student Package sections of the Maple Help System Table of Contents Limits To compute the limit of an expression as the independent variable approaches a value 1 In the Expression palette click the limit item ita 2 Specify the independent variable limit point and expression and then evaluate it Press Tab to move to the next placeholder For example x gt im x gt 0 sin x The limit Command By default Maple searches for the real bidirectional limit unless the limit point is 0 or 00 To specify a direction include one of the options left right real or complex in a call to the limit command See Table 5 8 176 5 Mathematical Problem Solving Table 5 8 Limits Command Syntax Output Using the limit command you can also compute multidimensional limits gt imi x L y For more information on multidim
130. The Student package is a collection of subpackages for teaching and learning mathematics and related subjects The Student package contains packages for a variety of subjects in cluding precalculus calculus and linear algebra Instructors can e Teach concepts without being distracted by the mechanics of the computations e Create examples and quickly update them during a lesson to demonstrate different cases or show the effect of the variation of a parameter e Create plots and animations to visually explain concepts for example the geometric re lationship between a mathematical function and its derivatives Tools Tutors Calculus Single Variable Derivatives See Figure 5 10 200 5 Mathematical Problem Solving I Calculus 1 Derivative Fie Help Plot Windows Enter a Function and an interval a b l Fis x costs a 0 b 2 Pj Derivatives i Pig costxi x sin x Display Fic in the plot l Pixi 2 sin x x cost x Display F isc in the plot Display Plot Options Maple Command S S ee a eee ES tat TE THOR eee ee 2 Piy order 1 view 0 e 6 60 4 24 7 24 Figure 5 10 Student Calculus1 Derivatives Tutor Students can e Perform step by step computations for example compute a derivative by applying dif ferentiation rules using commands or a tutor Tools Tutors Calculus Single Vari able Differentiation Methods See Figure 5 11
131. Y 10 Enter 3 and then press the Space bar gt 5 Evaluate 3x 42 Eag a dx FaN 1 l 11 Select the n th root symbol from the Ex n pression palette Va 12 Enter 3 then press Tab Drawing Plot Animal 13 Enter x then press Tab C 2D Math v _ Times New Roman x 7 5 eas Evaluate 3 re jx t ax dx Zy 1 14 Enter x for the integration variable 15 Click the Text icon in the toolbar then E math Drawing Plot Animation enter the rest of the sentence and write in C E text w Times NewRoman 2 7 BZU ay 5 Evaluate 3a 24 x 3y x dx and write in simplest terms A l simplest terms 1 3 Point and Click Interaction Maple contains many built in features that allow you to solve problems quickly without having to know any commands Assistants Maple offers a set of assistants in the form of graphical user interfaces to perform many tasks without the need to use any syntax An example of an assistant is shown in Figure 1 4 1 3 Point and Click Interaction 33 Solution loo jective Value 134 4911615397468162 x 4 5395592925391253 7 1 46440707460871 Figure 1 4 Optimization Assistant Using the Tools Assistants menu you can access tools to help you accomplish various tasks See Figure 1 5 In some cases you can launch an assistant by entering an expression and selectin
132. a Maple Command 4 Change the lower endpoint to P1 Select perivativerlot x cos x Pi Pi the check box to display J x in the plot Click Display to make these changes take effect 5 You can change the expression and modify plot options from within this tutor cos x For each change made click Display to Derivatives of Various Orders of view the altered plot When complete click f x x cos x on the Interval Pi Pi Close to display the resulting plot in the document The title and legend are automat ically added order 1 2 wiew 3 14 3 14 3 77 3 77 1 derivative tutor 5 8 Clickable Math Examples 217 Access the Tutor from a Task Template Maple also comes with a Task Template to solve this problem without using any commands 1 Launch the Task Template Browser by selecting 5 4 Calculus Differential Tools Tasks Browse ES Limits ee Derivatives 2 In the table of contents of the Task Browser Derivatives by Definition dialog select Calculus Differential Derivat Difference Newton Quotient j i ere i E Differentiation Formal Rul ives gt Graph f x and its Derivatives EG Peter a el ties PoE o Expression Functional Operator Graph Fis and Its Derivatives 3 Click Insert Minimal Content at the top of the dialog to insert the task template into the current Enter the function f x to be evaluated and the interval on
133. a dynamically generated menu of actions that are applicable for the region upon which it is invoked Context menus allow you to perform calculations and manipulations on expressions without using Maple syntax To display a context menu right click an object expression or region Context menus are available for many input regions including e expressions to perform calculations manipulations or plotting e plot regions to apply plot options and manipulate the plot e tables to modify the table properties e palette regions to add or remove palettes and palette regions e text regions to add annotations and format text 40 1 Getting Started e spreadsheets to manipulate the spreadsheet When performing calculations or manipulations on an expression a self documenting arrow or equal sign connects the input and output indicating the action that took place See Figures 1 8 and 1 9 for two examples of context menus e424 1 x41 Copy as MathML Numeric Formatting Explore pple a Command 45sign to a Mame Coefficients b i Collect Cut Cr Copy Ctrl C Complete Square d Copy Full precision Differentiate b Topy as MathMl Evaluate at a Point Paste Ctrlty Expand Style gt Line Factor Symbol b Point k Integrate b Line b ow Surface with Line ni Color b Surface Lirit Transparency b Surface with Contour SEQUENCE Glossiness Contour Series b Orientation Hidden Simplify b Lighting k Solve b Isolate Expression F
134. a list of items Change the items listed pz ListBox and enter code to execute when an item is selected peana Math Expression Enter or display a mathematical expres sion The value can be updated based on code in the docu ment or another embedded component Meter Select or display an integer or floating point value Change the display and enter code to execute when the value changes Plot Display a 2 D or 3 D plot or animation This plot or animation can be interacted with in the same way as other plots see Plots and Animations page 237 The value can be updated based on code in the document or another em bedded component You can also enter code to be executed when the execute code pointer is used to click or drag in the plot region mbedded Plot window Radio Button Use with other radio buttons to select one Cy RadioButton in a group Enter code to execute when the value changes 10 2 Using Embedded Components 391 Component Name and Description Inserted Image Rotary Gauge Select or display an integer or floating point value Change the display and enter code to execute when the value changes Slider Select or display an integer or floating point value Change the display and enter code to execute when the value changes Text Area Enter or display plain text The value can be updated based on code in the document or another embed ded component and you can enter code to execute when
135. ab toolbar icon is off Tab icon off Allows you to move between cells using the Tab key H Tab icon on Allows you to indent in the table using the Tab key H Tab between the cells of the table and enter the following expressions in the first column For each function from the context menu select Differentiate x Cut and paste the res ulting expression into the second column 5 x cos x we gt S5sin wx e 8 sin x cos x Modifying the Structural Layout of a Table The number of rows and columns in a table are modified using the Insert and Delete sub menus in the Table menu or by using the Cut and Copy Paste tools Inserting Rows and Columns Row and column insertion is relative to the table cell that currently contains the cursor If the document has an active selection insertion is relative to the selection boundaries 308 7 Creating Mathematical Documents e Column insertion can be to the left or right of the document position marker or selection e Row insertion can be above or below the marker or selection In your table add a third column on the right to display the plots of these expressions Add the heading and insert a blank plot region in each cell below it by selecting Insert Plot 2 D or 3 D for the second expression Then Ctrl drag Control drag for Macintosh each expression in the row into its plot region to display it For details on this procedure see Plots and Anim
136. able for polynomial operations Table 5 3 Select Other Polynomial Commands 158 5 Mathematical Problem Solving Command ecdex CurveFitting PolynomialInterpolation Interpolating polynomial for list of points See also the CurveFitting Assistant Tools Assistants Curve Fitting Least common multiple of two polynomials norm Norm EPROM randpoly Random polynomial PolynomialTools IsSelfReciprocal sqrfree Additional Information Table 5 4 Additional Polynomial Help General polynomial information polynom help page PolynomialTools package PolynomialTools package overview help page Algebraic manipulation of numeric polynomi SNAP Symbolic Numeric Algorithms for Polyno als mials package overview help page Efficient arithmetic for sparse polynomials SDMPolynom Sparse Distributed Multivariate Polynomial data structure help page Polynomial information and commands Maple Help System Table of Contents Mathemat ics Algebra Polynomials section 5 3 Linear Algebra 159 5 3 Linear Algebra Linear algebra operations act on Matrix and Vector data structures You can perform many linear algebra operations using task templates In the Task Browser Tools Tasks Browse expand the Linear Algebra folder Creating Matrices and Vectors Creating Matrices You can create a Matrix using e The Matrix command e The angle bracket shortcut notation e The Matrix palet
137. age 414 11 Input Output and Interacting with Other Products 11 3 Reading from Files The most common reason for reading files is to load data for example data generated in an experiment You can store data in a text file and then read it into Maple Reading Data from a File Import Data Assistant If you generate data outside Maple you can read it into Maple for further manipulation This data can be an image a sound file or columns of numbers in a text file You can easily import this external data into Maple using the Import Data Assistant where the supported file formats include files of type Excel MATLAB Image Audio Matrix Market and Delimited To launch the Import Data Assistant e From the Tools menu select Assistants and then Import Data e A dialog window appears where you can navigate to your data file Select the file that you want to import data from and then select the file type before clicking Next e From the main window you can preview the selected file and choose from the applicable options based on the format of the file read in before importing the data into Maple See Figure 11 1 for an example Data Import Assistant Additional Format Options Data Type integer 1 08 bit Skip Lines 2 Source From Rectangular ka F Transpose Separator Space Separated vi view of File Experiment 25 ae sl 2 Ame 3 d q Le 4 D da So Cancel Previous i Fig
138. age 126 The mod operator also supports polynomial and matrix arithmetic over finite rings and fields For more information refer to the mod help page Gaussian Integers Gaussian integers are complex numbers in which the real and imaginary parts are integers The GaussInt package contains commands that perform Gaussian integer operations The Glfactor command returns the Gaussian integer factorization 4 4 Solving Equations 111 gt GaussInt Glfactor 173 161 1 21 41 661 In Maple complex numbers are represented as at b I where the uppercase I represents the imaginary unit J 1 You can also enter the imaginary unit using the following two methods Inthe Common Symbols palette click the I i or j item See Palettes page 22 e Enter i or j and then press the symbol completion key See Symbol Names page 29 Note that the output will still be displayed with I no matter what symbol was used for input You can customize Maple s settings to use a different symbol for y 1 For more inform ation on entering complex numbers including how to customize this setting refer to the HowDol help page The GIsqrt command approximates the square root in the Gaussian integers gt GaussInt Glsqrt 9 5j a For more information on Gaussian integers including a list of GaussInt package commands refer to the GaussInt help page 4 4 Solving Equations You can solve a variety of equation types
139. ages and Tutors in an academic setting 149 150 5 Mathematical Problem Solving Clickable Math Examples page 213 Solve math Step by Step examples problems using some of the interactive methods available in Maple 5 2 Algebra Maple contains a variety of commands that perform integer operations such as factoring and modular arithmetic as described in Integer Operations page 106 In addition it supports polynomial algebra For information on matrix and vector algebra see Linear Algebra page 159 Polynomial Algebra A Maple polynomial is an expression in powers of an unknown Univariate polynomials are polynomials in one unknown for example X 2x 13 Multivariate polynomials are polynomials in multiple unknowns such as xy nf 7x The coefficients can be integers rational numbers irrational numbers floating point numbers complex numbers variables or a combination of these types gt a 5 i ax tie se Arithmetic The polynomial arithmetic operators are the standard Maple arithmetic operators excluding the division operator The division operator accepts polynomial arguments but does not perform polynomial division Polynomial division is an important operation The quo and rem commands find the quotient and remainder of a polynomial division See Table 5 1 The iquo and irem commands find the quotient and remainder of an integer division For more information see Intege
140. al Equations tutor DE Plots is accessible through the DEtools package For a definition of the term package see Package Commands page 47 The Student package is a collection of subpackages designed to assist with the teaching and learning of standard undergraduate mathematics The subpackages contain many com mands for displaying functions computations and theorems in various ways and include support for stepping through important computations The interactive commands help you explore concepts and solve problems using a point and click interface These commands launch tutors that provide a graphical interface to some of the visualization and computation commands described above See Figure 1 7 for an example of one of the tutors 1 3 Point and Click Interaction 39 W Calculus 1 Differentiation Methods Fie Edit Rule Definition Apply Rule Understood Rules Help Enter a Function Function ty ecostcifsin x Variable x d cas x Y dx sin x d d of cos x 2x E E dx sial x cas x sig x cio sir xX tla SiR X sin x cas x siv X sin X COs X Power e Chain Rule E one A Integral Rewrite Exponential Matural Logarithm t lt hyperbolic gt i Show Hints Get Hint lt arctrig gt ow lt carchyperbolic gt Figure 1 7 Calculus Single Variable Differentiation Methods Tutor Context Menus A context menu is
141. al Content Insert into New Worksheet w Integration Display task markers e Approximate Integration os T Approximate Definite Integral of a Function es T Numeric Integration Appr oximate Definite E E Methods of Integration H I Applications E E Series H P Calculus Multivariate ie 4 Calculus vector i eT Convert Expression to Function B i Curve Fitting H 0 Differential Equations H E Document Templates H 6 Evaluating is 0 Geometry Integers E 0 Linear Algebra ie E Lists H 0 Maple T A E E Plots E E Polynomials E E Statistics E Transformations co Units Constants and Errors Task ApproxDefIntegralUnivariateFcn Figure 1 10 Browse Tasks Dialog Integral of a Function Description Approximate the definite integral of a univariate function using a Riemann sum or a Newton Cotes method hi the function as an expression 2 gt x 2 2 1 Specify the range of integrati on and the method of approximation and then approximate the integral gt Studenti Calculus Approximatelnt If 1 25 method an me err i 4 Prit iili 42 e 1 Getting Started Previewing Tasks To preview Maple tasks e From the Tools menu select Tasks and then Browse The Browse Tasks dialog opens and displays the list of tasks The tasks are sorted by subject to help you quickly find the desired task In the Browse Tasks dialog you can view tasks without inserting
142. ample 1 Using Context Menus Determine the rational expression fraction that approximates the floating point number 0 3463678 1 7643 1 Enter and execute the expression gt 0 3463678 1 7643 2 1106678 2 Right click Control click for Macintosh the 2 1106673 output floating point number Copy as MathML Numeric Formatting Explore Apply a Command Assign to a Mame Next Float Previous Float Conversions b Continued Fraction Integer Functions p Exact Rational Units b Rational 3 From the context menu select Conversions gt gt conve rt 3 3 rationat Rational The inserted calling sequence includes an equation label reference to the number you are converting Notice that an equation label reference has been used For information on equation labels and equation label references see Equation Labels page 97 For more information on context menus see Context Menus page 70 in Chapter 2 3 6 Assistants and Tutors 91 3 6 Assistants and Tutors Assistants and tutors provide point and click interfaces with buttons text input regions and sliders See Figure 3 3 ODE Analyzer Assistant a Differential Equations Conditions Parameters Solve Numerically Solve Symbolically Classify Figure 3 3 ODE Analyzer Assistant Launching an Assistant or Tutor To launch an assistant or tutor 1 Open the Tools menu 2 Select Assistants or Tutors 3 Navigate
143. anchor Resizing arrows appear 3 Click and drag the image to the desired size Note To constrain the proportions of the image as it 1s resized press and hold the Shift key as you drag You can also draw on images in the same way as the drawing canvas For more information refer to the worksheet documenting drawingtools help page ImageTools Package You can manipulate image data using the ImageTools package This package 1s a collection of utilities for reading and writing common image file formats and for performing basic image processing operations within Maple Within Maple images are represented as dense rectangular Arrays of 64 bit hardware floating point numbers Grayscale images are 2 D whereas color images are 3 D the third dimension representing the color channels In addition to the commands in the ImageTools package many ordinary Array and Matrix operations are useful for image processing For details about this feature refer to the ImageTools help page 7 6 Hyperlinks Use a hyperlink in your document to access any of the following e Web Page URL e Email e Worksheet e Help Topic e Task e Dictionary Topic e Maplet 324 7 Creating Mathematical Documents Hyperlink Properties Link Text Type Help Topic Target Bookmark Figure 7 18 Hyperlink Properties Dialog Inserting a Hyperlink in a Document To create a hyperlink from exist
144. and are a product not a single entity The following calling sequences define different expressions gt 1 Unit m 2 Unit s gt 1 Unit m 2 Unit s m s gt m Es l gt ae Some units support prefixes For example SI units support prefixes to names and symbols You can specify 1000 meters using kilometer or km For more information refer to the Units prefixes help page gt 1 5km 1 5 km Performing Computations with Units In the default Maple environment you cannot perform computations with quantities that have units You can perform only unit conversions For more information about the default environment refer to the Units default help page To compute with expressions that have units you must load a Units environment Natural or Standard It is recommended that you use the Standard environment gt with Units Standard In the Standard Units environment commands that support expressions with units return results with the correct units 4 5 Units Scientific Constants and Uncertainty 133 gt area 3 ft mite 14370939 gt 78125 Lm area 7 12 sin x x m s I 12sin x 27 4 15 gt int 4 15 x s 12 cos x el n 4 16 gt diff 4 16 x s 12sin 27 For information on differentiation and integration see Calculus page 175 Changing the Current System of Units If a computation include
145. and Side To extract the left hand side of an equation inequality or range e Use the Ihs command To extract the right hand side of an equation inequality or range e Use the rhs command 8 3 Working with Maple Expressions 351 For example gt y x 1 y x 1 8 1 gt lhs 8 1 y 8 2 gt rhs 8 1 x 8 3 For the following equation the left endpoint of the range is the left hand side of the right hand side of the equation gt x 3 5 x 3 5 8 4 gt Ihs rhs 8 4 ad 8 5 Numerator and Denominator To extract the numerator of an expression e Use the numer command To extract the denominator of an expression e Use the denom command 1 sin x H lt gt g y I x If the expression is not in normal form Maple normalizes the expression before selecting the numerator or denominator For more information on normal form refer to the normal help page 352 e 8 Maple Expressions gt numer e x sin x ae gt denom e x 4 x gt denom denon e The expression can be any algebraic expression For information on the behavior for non rational expressions refer to the numer help page Components of an Expression The components of an expression are called its operands To count the number of operands in an expression e Use the nops command For example construct a list of solutions to an equation gt solutions solve 6x
146. and use basic data structures Gai Lists Tables Arrays Matrices and Vectors Functional Operators Strings Working with Maple Expressions page 348 Tools Low Level Operations for manipulating and controlling the evaluation of expressions Manipulating Expressions Evaluating Expressions 8 2 Creating and Using Data Structures Constants data structures mathematical expressions and other objects are Maple expressions For more information on expressions refer to the Maple Help System This section describes the key data structures e Expression sequences e Sets e Lists e Tables e Arrays e Matrices and Vectors e Functional operators e Strings 337 338 8 Maple Expressions Expression Sequences The fundamental Maple data structure is the expression sequence It is a group of expressions separated by commas gt 2y sin x Accessing Elements To access one of the expressions e Enter the sequence name followed by the position of the expression enclosed in brackets For example gt S 2 Using negative integers you can select an expression from the end of a sequence gt S 2 sin a You can select multiple expressions by specifying a range using the range operator gt 2 2 y sin x Note This syntax is valid for most data structures Sets A set is an expression sequence enclosed in curly braces l 2 gt 44 12i sin
147. ane Hurnicolurins 3 numo reference BosRow2 a In the drop down list select BoxRow2 valign b Change the numcolumns field to 3 Add a label to row 2 wl EH Ee 7 1 4 From the Body palette drag the Label i me 44 ea ee element to the left column in the Layout erui pane F z i L Sae Label1 tooltip a In the drop down list select Labell z _ ToolBar 5 In the Properties pane TNE vil tne Enter a function of x b Change the caption field to Enter a function of x 406 10 Embedded Components and Maplets Add a text region to row 2 6 From the Body palette drag the Text Field element to the middle column The TextField element allows the Maplet user to enter input that can be retrieved in an action 7 If necessary resize the Maplet Builder to display the entire Layout pane Add a button to row 2 8 From the Body palette drag the Button ai element to the right column in the Layout pane 9 In the Properties pane a In the drop down list select Button1 b Change the caption field to Plot c In the onclick property drop down list select lt Evaluate gt ile 4 64 ae Y Dialog ely hes KE KE Y ToolBar Enter a function of x _ Dialog IFEF aoo V Menu P EBEERI _ ToolBar
148. aple commands needed to produce the solution values and the displayed plot 4 4 Solving Equations 125 For more information refer to the 0DEAnalyzer help page The dsolve Command The ODE Analyzer provides a point and click interface to the Maple dsolve command For ODEs or systems of ODEs the dsolve command can find e Closed form solutions e Numerical solutions e Series solutions In addition the dsolve command can find e Formal power series solutions to linear ODEs with polynomial coefficients e Formal solutions to linear ODEs with polynomial coefficients To access all available functionality use the dsolve command directly For more information refer to the dsolve help page Partial Differential Equations PDEs To solve a PDE or PDE system symbolically or numerically use the pdsolve command PDE systems can contain ODEs algebraic equations and inequations For example solve the following PDE symbolically For help entering a partial derivative see Example I Enter a Partial Derivative page 65 gt gren assy 0 0 f 56A gran 0 4 14 gt pdsolve 4 14 f x y Fi Ea y The solution is an arbitrary univariate function applied to x y l Maple generally prints only the return value errors and warnings during a computation To print information about the techniques Maple uses increase the infolevel setting for the command To return all information set infolevel to
149. ary topic adding hyperlink to 326 diff command 122 178 differential equations ordinary 121 partial 125 differentiation 177 with uncertainty 141 with units 133 Differentiation Methods Tutor 200 Digits environment variable 105 dimension 128 171 base 128 Directional Derivative Tutor 180 discrim command 158 display bookmark 327 distribution probability 193 divide command 152 divisors 108 document blocks 51 300 Document mode 63 documents running 9 DocumentTools 398 double colon operator 144 dsolve command 125 E e notation 104 Edit menu in help system 57 eigenvalues 171 eigenvectors 171 element wise operators 363 elementary charge 135 elements 134 definition 136 isotopes 136 definition 136 properties 136 list 136 properties list 136 uncertainty 138 units 138 using 136 value 137 value and units 138 elif clauses 371 order 372 else clause 371 email adding hyperlink to 325 embedded components 329 389 inserting 393 properties 393 end do keywords 373 376 377 end if keywords 370 end proc keywords 383 engineers portal for 58 environment variables Digits 105 Order 182 equation solving step by step 218 equation labels 99 displaying 97 features 100 formatting 51 inserting 50 numbering schemes 99 overview 49 references to 97 versus names 100 with multiple outputs 99 Equation Manipulator 36 218 equations solving 112 for real so
150. ate and vector calculus operations on VectorCalculus vectors vectors with an additional coordinate system attribute and vector fields vectors with additional coordinate system and vectorfield attrib utes for example Curl Flux and Torsion gt with VectorCalculus gt BasisFormat false gt SetCoordinates cartesian x y z gt VectorField VectorField y x 2 VectorFieldl x Note For information on changing the display format in the VectorCalculus package see the VectorCalculus BasisFormat help page 5 4 Calculus 187 Find the curl of VectorField1 gt Curl VectorField 0 0 Find the flux of VectorField1 through a sphere of radius r at the origin gt Flux VectorField Sphere 0 0 0 7r Compute the torsion of a space curve The curve must be a vector with parametric function components gt simplifv Torsion t K P t assuming t real 9f 9P 1 For information on the assuming command see The assuming Command page 146 For more information on the VectorCalculus package including a complete list of com mands refer to the VectorCalculus help page To find other calculus packages such as VariationalCalculus refer to the index package help page Student Calculus Packages The Student package contains subpackages that help instructors teach concepts and allow students to visualize and explore ideas These subpackages also contain computational comman
151. ation 189 IH Optimization Assistant x a x QUE Objective Function lx yee gt Nonlinear x 0 5 s epee O Minimize Maximize Foasidity Tolerance defaut Objective valus 1394 491161539748162 Optimality Tolerance def auk x 53559292539129 Y 1 46440707460871 Ror tion Limit defa infinite Bound def aul On Quit Return Solution Figure 5 8 Optimization Assistant To solve a problem 1 Enter the objective function constraints and bounds 2 Select the Minimize or Maximize radio button 3 Click the Solve button The solution is displayed in the Solution text box You can also enter the problem objective function constraints and bounds in the calling sequence of the Optimization Interactive command For example find the maximum value of xy y subject to the constraints x y 627 OS 7 05 gt Optimization Interactive x y y x y lt 6 x 0 5 y 0 5 134 491 161539748162 x 4 53559292539129189 y 1 46440707460870746 e When the Optimization Assistant opens select Maximize then Solve 190 5 Mathematical Problem Solving After finding a solution you can plot it To plot a solution e Inthe Optimization Assistant window click the Plot button The Optimization Plotter window is displayed See Figure 5 9 Note When you close the Optimization Assistant you can choose to return the solution problem command used plot or nothing using the drop down in the
152. ations page 237 Resize the plots and table as desired 5x cos mx we 5sin x e 8 sin x cos x Deleting Rows and Columns With deleting operations using the Delete key the Delete Table Contents dialog opens allowing you to specify the desired behavior For example you can delete the selected rows or delete the contents of the selected cells See Figure 7 11 7 4 Tables 309 Delete Table Contents Delete Cell Contents Figure 7 11 Delete Table Contents Verification Dialog Pasting Pasting a table subselection into a table may result in the creation of additional rows or columns overwriting existing cell content or the insertion of a subtable within the active table cell When there is a choice the Table Paste Mode dialog opens allowing you to choose See Figure 7 12 Table Paste Mode E9 Replace cell contents Figure 7 12 Table Paste Mode Selection Dialog Merging Cells To merge adjacent cells in a table select the cells you would like to merge From the Table menu select Merge Cells You can merge cells across row or column borders See Figure 7 13 The resulting cell must be rectangular The contents of the individual cells in the merge operation are concatenated in execution order See Figure 7 14 For details on cell execution order see Execution Order Dependency page 315 Figure 7 13 Two cells Figure 7 14 Merged Cells 310 7 Creating Mathematical Do
153. aying expres sions The rule based typesetting functionality is available when the Typesetting level is set to Extended Tools Options Display tab This parsing functionality applies to 2 D Math editing Math mode only For example you can change the display of derivatives to suit the content and audience of your document 304 7 Creating Mathematical Documents gt at Tools Options Display tab Typesetting level Maple Standard f x de gt a x dx Tools Options Display tab Typesetting level Ex ren tended F x To specify rules use the Typesetting Rule Assistant e From the View menu select Typesetting Rules The Typesetting Rule Assistant dialog opens For more information see the Typesetting TypesettingRuleAssist and OptionsDia logDisplay help pages Auto Execute The Autoexecute feature allows you to designate regions of a document for automatic exe cution These regions are executed when the document opens or when the command is ex ecuted This is useful when sharing documents Important commands can be executed as soon as the user opens your document The user is not required to execute all commands Setting the Auto Execute Feature 1 Select the region to be automatically executed when the document opens 2 From the Format menu select Autoexecute and then Set Regions set to Autoexecute are denoted by exclamation mark symbols in the Markers region View Markers
154. ble representation of Function and derivatives DETools df table i expr diffalg ciffalz diffop2de differential operator DFTools diffonzde loper y iw gt 2 Select the partial differentiation item i diff inline partial a7 3 Replace the placeholder with t Use the right arrow to move out of the denominat 2 al l or Enter as in the previous ex ample Example 2 Define a Mathematical Function Define the function twice which doubles its input 2 4 Evaluating Expressions 67 1 In the Expression palette click the single variable Tay function definition item a 2 Replace the placeholder f with the function name hpiee F y twice Press Tab to move to the next placeholder 3 Replace the parameter placeholder a with the inde paseo m y J pendent variable x Press Tab 4 Replace the output placeholder y with the desired twice x gt 2x twice 1342 2684 twice y z 2y 2z Note To insert the right arrow symbol you can also enter the characters gt in Math mode In this case symbol completion is automatic Important The expression 2 x is different from the function x gt 2x For more information on functions see Functional Operators page 343 2 4 Evaluating Expressions To evaluate a mathematical expression place the cursor in the expression and press Ctrl Command for Macintosh That 1s press and hold the Ctrl or Command
155. box solidcircle or soliddiamond for 2 D plots asterisk box circle cross diagonalcross diamond point solidsphere or sphere for 3 D plots Defines a title for the plot Defines the thickness of lines in the plot transparency 3 D Controls the transparency of the plot surface 268 6 Plots and Animations view Defines the minimum and maximum coordinate values of the axes displayed on the screen For a complete list of plot options refer to the plot options and plot3d options help pages gt plot Si x x 20 20 title Plot of the Sine Integral titlefont HELVETICA 12 color hlue style point Plot of the Sine Integral amp gt F 6 4 Analyzing Plots 269 To create a smoother or more precise plot calculate more points using the numpoints option xy 244A gt plot3d x 10 10 y 10 10 aves hoxed numpoints 1500 lightmodel light3 X shading zgrayscale orientation 160 20 style patchnogrid 6 4 Analyzing Plots Point Probe Rotate Pan and Zoom Tools To gain further insight into a plot Maple offers various tools to analyze plot regions These tools are available in the Plot menu menu Context Bar and in the context menu under Transform when the plot region is selected Table 6 4 Plot Analysis Options Point probe r Display the coordinates corresponding to the cursor position on a two di 2 D mensional plot in the context bar upper left
156. ccess it The two important differences are e The indices can be any integers e The dimension can be greater than one Creating and Using Arrays To define an Array use the Array constructor Standard Array constructor arguments are e Expression sequences of ranges Specify the indices for each dimension e Nested lists Specify the contents For example gt a Array 1 3 1 3 1 1 2 3 4 5 6 17 8 911 S amp DO A i CA Wa gt b Array 1 2 2 5 1 2 4 9 6 3 7 1 9 2 5 5 2 4 1 7 b Array 2 2 5 1 2 1 2 1 3 439 1 4 63 1 5 72 4 2 92 2 3 5 5 2 4 2 4 2 5 1 7 datatype anything storage rectangular order Fortran order To access entries in an Array use either square bracket or round bracket notation 8 2 Creating and Using Data Structures 341 Square bracket notation respects the actual index of an Array even when the index does not start at 1 gt all 1 gt al2 3 6 gt bf2 3 Lan Ci gt b 1 1 Error Array index out of range Round bracket indexing normalizes the dimensions to begin at 1 Since this method 1s relative you can access the end of the array by entering 1 gt a 1 2 gt b 1 1 bed The Array constructor supports other syntaxes It also supports many options For more information on the Array constructor and the Array data structure refer to the Array help page For more informatio
157. ce to manipulate the libraries in a specified directory Maplet Builder an interface to the Maplets package The Maplets package contains commands for creating and displaying Maplet applications point and click interfaces Using the Maplet Builder you can define the layout of a Maplet drag and drop elements visual and functional components of Maplets set actions associated with elements and directly run a Maplet application The Maplet Builder is available in the Standard interface only ODE Analyzer an interface to obtain numeric or symbolic solutions to a single ordinary differential equation ODE or a system of ODEs and plot a solution of the result Optimization an interface to the solver commands in the Optimization package The Optimization package is a collection of commands for numerically solving optimization problems which involves finding the minimum or maximum of an objective function possibly subject to constraints Plot Builder an interface for creating two and three dimensional plots animations and interactive plots Scientific Constants an interface to over 20 000 values of physical constants and properties of chemical elements All of these constants come with the corresponding unit and if applicable with the uncertainty or error that is how precisely the value of this constant is known Special Functions an interface to the properties of over 200 special functions including the Hypergeometric B
158. cetyesivantseeesonadsiouwnameotonsdives 198 Student Packages and TUtOrS ssicsssrcrsissrciecasreresirsis diiri ta rera rer EE A EEIN ARa 199 Calculus Problem Solving Examples cccccccceeceseeceeseeseseaeeaeeseeens 206 5 0 Clickable Math Examples comisncocensincaensedsateiminsbauecssnseiinbecdniaponiiaaucerunaans 213 Example 1 Graph a Function and its Derivatives ccccceceeeeeeneeeeeeeees 213 Example 2 Solve for x in a Quadratic Equation cccceeeseeceseeeeees 218 Example 3 Solve a Quadratic Trig Equation ccccccccec ees eeeeeeeneesenees 223 Example 4 Find the Inverse Function sisecscndsesineadiscaewsSntnsedyedvandosduececatedsien s 220 vi Contents Example 5 Methods of Integration Trig Substitution ccce eee ees 230 Example 6 Initial Value Problem isjsiacecccnssencnnntneesondeaiemiiessssinwennvererie ceiensecs 233 6 Plots and A ONG os oes esas scarce picts ANTENA eon A EENEN ENERE cs 237 oe bigs Wario Och oli eee ne een E meee enna ne oon eee ee eee 231 Oe eat a eee E AE EI EE N E E E E 238 lnteracive Plo Dulder srransrronisrr nrnna ERE EEE EE EEEE 238 COn T M a E E E cree uted 246 Dracena toa PIOC RETION aepecpcsaeeancascnseeeanectansessscesennoueaecteasea caren 249 The plot and plot3d Commands gccsixcicenvcaeceseveseeedersenweteeeeddercesenedeacenvenses 249 Tapon PaE ae reren S EE EE EE T EEE 257 Multiple Plots in the Same Plot ReG 100s c0nsecvc
159. command gt k x gt sin cos x z say E gt in K 0 1 0 gt r StruveH 0 x gt i t For information on integration and other calculus operations see Calculus page 175 Strings A string is a sequence of characters enclosed in double quotes gt S This is a sequence of characters Accessing Characters You can access characters in a string using brackets gt S 11 2 sequence of characters Using Strings The StringTools package is an advanced set of tools for manipulating and using strings gt with StringTools 348 8 Maple Expressions gt Random 9 alnum 8dvpw bJm gt Stem impressive impress gt Split Create a list of strings from the words in a string ae i o Nn A on Create a list of strings from the words in a string 8 3 Working with Maple Expressions This section describes how to manipulate expressions using commands Topics covered include testing the expression type accessing operands of an expression and evaluating an expression Low Level Operations Expression Types A Maple type is a broad class of expressions that share common properties Maple contains over 200 types including CF e boolean e constant e integer e Matrix e trig e truefalse For more information and a complete list of Maple types refer to the type help page The type commands return true if the expression satisfie
160. crants 385 Displaying Maple Library Procedure Definitions ccc ccc ccee ee eeeen sees 385 WI OCICS erreren ee EEEE EE E E deep ener EE EEEE 386 9 5 Programming in OCMC IS ssirisecririssciirribdr trind d irienn niis Eiee NTETE KEN 387 Code Edit RELION ssisrorrsicrikrei orit hEn na rA EAE RE A TEE 387 PAI COG aep pace ES E S E E E E EE EEE E sant 388 Bo OM A BIOK corrien ear E E EE EE ae ae E EE 388 10 Embedded Components and Maplets ois s0sceccswe cunnwecsine reese tavninccenponseodianteneegenenies 389 10 1 To Tus Chaple pesrseaece rh rcnerasoentaesaeontovies nose E ERE EEEE EEEE 389 10 2 Using Embedded Components lt lt csnscnesausteccecsnssnosmanagressasesranceeaeseteeunenvas 389 EET E E PEE E EI A NAE EEE E EE E O 389 Printing and Exporting a Document with Embedded Components 392 10 3 Creating Embedded Components ci ccenatscetacxcaeescencecawnscosndensedatuesadanacnance 393 TASC REINO COmMpONCNIS sonnsaceceesseancesecaceeccesaentoncnnuscsoaeuiecacusnnotencatoouasees 393 Editing Component Properties General Process siinsciacscasiadsvevantesentebatecceiods 393 Removing Graphical Interface Components ccc ececeec eee eeeeeeeseneenees 394 Integrating Components into a Document ccc ccc cc ccc ec ence nena eee eeseeenes 394 Example 2 Creating Embedded Components ccccceceeeeaeeceeeeneenes 396 OA Usina INVADICES speren E N asians EEEE EREE 400 KPEE FE erra E AE
161. create custom character styles to apply to text or change existing character styles New styles are automatically added to the styles drop down list in the context bar of your document 1 From the Format menu select Styles The Style Management dialog opens See Figure 7 4 To create a character style e Click Create Character Style The Character Style dialog opens See Figure 7 5 e In the first row of the dialog enter a style name in the blank text region To modify a character style e From the style list select the character style to modify Recall that all character styles are preceded by the letter C while paragraph styles are preceded by the letter P e Click Modify The Character Style dialog opens with the current attributes displayed See Figure 7 5 For either action continue 2 Select the properties for the new character style such as font size attributes and color In the font attributes the Superscript and Subscript check boxes are mutually exclusive When you select one of the two check boxes the other is disabled You must clear one before selecting the other Note A preview of the style is displayed in the last row of the Character Style dialog 3 To save the style click OK or to abandon click Cancel If you have modified a style all text in your document that uses the altered style is updated to reflect the changes 7 2 Document Formatting 291 wc haracter Style ee g EN E th ore
162. cuments Modifying the Physical Dimensions of a Table The overall width of the table can be controlled in several ways The most direct way is to press the left mouse button press mouse button for Macintosh while hovering over the left or right table boundary and dragging the mouse left or right Upon release of the mouse button the table boundary is updated This approach can also be used to resize the relative width of table columns Alternatively the size of the table can be controlled from the Table Properties dialog Select the Table menu and then Properties Two sizing modes are supported 1 Fixed percentage of page width Using this option the table width adjusts whenever the width of the document changes This option is useful for ensuring that the entire content of the table fits in the screen or printed page 2 Scale with zoom factor This option is used to preserve the size and layout of the table regardless of the size of the document window or the zoom factor If the table exceeds the width of the document window the horizontal scroll bar can be used to view the rightmost columns Note Using this option tables may be incomplete when printed Modifying the Appearance of a Table Table Borders The style of exterior and interior borders 1s set using the Table Properties dialog From the Table menu select Properties e You can set all none or only some of the borders to be visible in a table Exterior borders
163. den in a table even if they are set to visible for the document in the View Show Hide Contents dialog 7 4 Tables 315 Printing Options The Table Properties dialog contains options to control the placement of page breaks when printing You can fit a table on a single page allow page breaks between rows or allow page breaks within a row Execution Order Dependency The order in which cells are executed is set in the Table Properties dialog The following tables illustrate the effect of execution order Row wise execution order gt x x 1 gt x x 1 Tables and the Classic Worksheet Tables are flattened on export to the Classic Worksheet interface For example the following table in the Standard Worksheet appears as one column in the Classic Worksheet interface 316 7 Creating Mathematical Documents Additional Examples For more practice creating and manipulating tables try creating the following tables at the end of your document Table of Values This example illustrates how to set the visibility options for cell contents to display a table of values 9 gt p tot 5 Create a table with 2 rows and 7 columns Enter the values as below and then select all table cells In the Table Alignment menu select Columns and then Center cases sa n Table settings In the Properties dialog Table Properties menu 1 Set Table Size Mode to Scale with zoom factor 2 Hide Mapl
164. dialog Enter the label number in the Insert Label dialog and click OK The item is now a label See Figure 1 12 sin x dx ia K cos x D Hire Insert Label Type Equation Identifier 1 Figure 1 12 Inserting an Equation Label 4 Press Enter to obtain the result To change the format of equation labels e Select Format Labels Label Display In the Format Labels dialog select one of the numbering schemes e Optionally enter an appropriate numbering prefix 1 4 Commands 51 i cosl x Answerl db z Answerl dx p sil x Answer Format Labels Label Numbering Prefix Answer Label Numbering Scheme Flak Mumeric w Figure 1 13 Format Labels Dialog Adding a Prefix The Label Reference menu item allows you to switch between the label name and its ref erence content Place the cursor on the referenced equation label and select Format Labels Label Reference sini x 2 Figure 1 14 Label Reference The label is associated with the last output within an execution group You cannot apply equation labels to the following e Error warning and information messages e Tables images plots sketches or spreadsheets Document Blocks In Document mode content is created as a series of document blocks Document blocks allow you to hide the syntax used to perform calculations which in turn lets you focus on the concept presented instead of the command used t
165. dif z x x xy y 2z y 7x z x y 272 2y 0250 8 9 You can use values in one data structure type to compute values in another data structure type as long as both data structures are dimensional and contain the same number of ele ments In the following example the values in an Array are compared to the values in a Matrix that contains the same number of elements 8 3 Working with Maple Expressions 365 gt 12 88 20 gt 3 100 25 lt 2 100 lt 88 8 10 25 lt 20 For more information refer to the elementwise help page Levels of Evaluation In a symbolic mathematics program such as Maple you encounter the issue of levels of evaluation If you assign y to x z to y and then 5 to z what is the value of x At the top level Maple fully evaluates names That is Maple checks if the name or symbol has an assigned value If it has a value Maple substitutes the value for the name If this value has an assigned value Maple performs a substitution recursively until no more substitutions are possible For example gt x y gt yl Z gt z 5 Maple fully evaluates the name x and returns the value 5 gt xX To control the level of evaluation of an expression e Use the eval command with an integer second argument If passed a single argument the eval command fully evaluates that expression If you specify an integer second argument Maple evaluates the expression to that level 366 8
166. document block 1 Place the cursor in the document block region 2 From the View menu select Expand Document Block hal Plot the expression sin x and its integral gt n x dx f gt print 1 input placeholder oultpui redirected cos x A l in the same plot 7 3 Commands in Documents 303 3 To hide code again select View and then Collapse Document Block Expand an Execution Group within a Document Block An execution group is a grouping of Maple input with its corresponding Maple output It is distinguished by a large square bracket at the left called a group boundary As document blocks can contain many execution groups you can select to expand an exe cution group within a document block 1 Place the cursor near the end of the document block region 2 From the View menu select Expand Execution Group prj Plot the expression sin x and its integral sin x dx cos x A l in the same plot 3 To hide the group select View and then Collapse Execution Group Switch between Input and Output 1 Place the cursor in the document block region 2 From the View menu select Toggle Input Output Display Input from any executable math or commands is displayed in one instance or only output is displayed Typesetting You can control typesetting and 2 D Math equation parsing options in the Standard Work sheet interface Extended typesetting uses a customizable set of rules for displ
167. ds The Student calculus subpackages include Calculus1 MultivariateCalculus and VectorCalculus The Student VectorCalculus package provides a simple interface to a limited subset of the functionality available in the VectorCalculus package For information on using Maple as a teaching and learning tool and some computational examples see Teaching and Learning with Maple page 198 188 5 Mathematical Problem Solving 5 5 Optimization Using the Optimization package you can numerically solve optimization problems The package uses fast Numerical Algorithms Group NAG algorithms to minimize or maximize an objective function The Optimization package solves constrained and unconstrained problems e Linear programs e Quadratic programs e Nonlinear programs e Linear and nonlinear least squares problems The Optimization package contains local solvers In addition for univariate finitely bounded nonlinear programs with no other constraints you can compute global solutions using the NLPSolve command To find global solutions generally purchase the Global Optimization Toolbox For more information visit http www maplesoft com products toolboxes Point and Click Interface The primary method for solving optimization problems is the Optimization Assistant To launch the Optimization Assistant e From the Tools menu select Assistants and then Optimization Maple launches the Optimization Assistant See Figure 5 8 5 5 Optimiz
168. e Sorting Terms To sort the terms of a polynomial use the sort command 5 2 Algebra 153 gt pla x x ee a E pi x x ers gt sort pl Y x4 x2 x Note The sort command returns the sorted polynomial and updates the order of the terms in the polynomial The terms of p1 are sorted gt pl x 2 x To specify the unknowns of the polynomial and their ordering include a list of names 2 3 J gt sort a x tx xatatb a 2 r o i Xa xat at x b gt sort dx ir xatat b x b dy f tartia By default the sort command sorts a polynomial by decreasing total degree of the terms gt pra y y gt sort p2 x y x y x p The first term has total degree 4 The other two terms have total degree 3 The order of the final two terms is determined by the order of their names in the list To sort the terms by pure lexicographic order that 1s first by decreasing order of the first unknown in the list option and then by decreasing order of the next unknown in the list option specify the plex option 154 e 5 Mathematical Problem Solving gt sort p2 x y plex x J f J j For information on enclosing keywords in right single quotes see Delaying Evalu ation page 366 The first term contains x to the power 3 the second x to the power 2 and the third x to the power 0 Using context menus you can perform operations such as sorting for polynomials and many other Map
169. e x gt x Constants T A Pee Mathematical functions sin x cos wa Names variables x y z B Data structures sets lists Arrays Vectors Matrices Maple contains over a thousand symbols For some numbers operators and names you can press the corresponding key for example 9 gt or x Most symbols are not available on the keyboard but you can insert them easily using two methods palettes and symbol names Example 1 Enter a Partial Derivative To insert a symbol you can use palettes or symbol names o O l Enter the partial derivative mi i using palettes 66 2 Document Mode 1 In the Expression palette click the partial differ entiation item x f Maple inserts the partial deriv ative The variable placeholder is selected 2 Enter t and then press Tab The expression placeholder is selected _ 72 3 Enter Note To enter the exponential e use the expression palette or command completion To evaluate the integral and display the result inline press Ctrl Command for Macintosh or Enter For more information see Computing with Palettes page 69 You can enter any expression using symbol names and the symbol completion list 1 Begin typing the name of the symbol diff and press the symbol completion key see Shortcut Keys by Plat form page xiv diff_table Function and derivatives POFfoals d fable exor diff _ta
170. e inked to Ei evious List iter Initial List Value Bullet Suffix Page Break Before Linebreak Space iw Figure 7 6 Paragraph Style Dialog Style Set Management Saving Styles for Future Use You can use the style set of a particular document as the default style for all documents 7 2 Document Formatting 295 Style Set Management Current Style Set User defined Style Set Style Set Operations Revert to Style Set Apply style definitions From the current style set Apply Style Set Load style definitions From another worksheet New Style Set Create a new style set file Figure 7 7 Style Set Management Dialog For information on creating and managing style sets see the worksheet documenting styles help page Sections You can organize your document into sections either before or after the text has been entered First Section The introductory sentence gt cose x Subsection gt sin dx Using the Insert Menu to Add Sections 1 Place the cursor in the paragraph or execution group above the location at which you want to insert a new Section e Ifthe cursor is inside a section Maple inserts the new section after the current section e If the cursor is in an execution group Maple inserts the new section after the execution group 296 e 7 Creating Mathematical Documents 2 From the Insert menu select Section An arrow marks the start of the section 3 Enter th
171. e For this 0 397745 example enter the data 0 707107 as shown 1 104854 Current Page 1 Previous Page Next Page 3 Once you have BE Curve Fitting Assistant entered the data click the Fit button The second dialog of the Curve Fitting Assist lea Vertical range 1096700 J000e 1 ant appears Horizontal range 27 20 127 20 Per Ports Plot Points Polynomial Interpolation Splines Degree of the spline 1 Least Squares Enter an expression inx latx b re CEt Forts on tree epi 36 1 Getting Started Action Result in Document 4 In this dialog you 3 Curve Fitting Assistant can plot the data and several types of inter i i i i i ne Faul polations including Horizontal range 27 20 127 20 vertical range 1096700 3000e 1 Polynomial Spline and Least Squares For example click the Sette core Plot button in the Polynomial Interpola tion section The poly nomial is plotted with the data and the inter Least Squares polating function is Enter an expression in x a x b displayed below Splines Degree of the spline 2762122808e 2 x 7 4971820957e 1 x 6 3259304755 x 5 8949276991 x 4 665671371 x 3 9888404085 x 2 1 082752470 x 44194e 1 On Done retur ters 5 You can choose to return either the inter polating function or the plot to your document When finished click Done 2182341270 10 x 4 027388889 108 x
172. e pana variables optional nase of cae nonna for which io solve Comp lem optional beard noms search for complex sodotons Basic Information Y Description The mive command pamenically computes the raees of one of more equations exapreveons of procedures Output a The lanes te d migle Ue We iced nn pee inqoence The solutions te a set or hist of equations are retarned an sets of equanen teqeences For a aagle polynomial equations of one vanable with real coefficients by defuit the bive command computes af real non complen roots It may mA return all roots for exceptionally i conditioned polynomial For a dagie polynomial equation of one vanable with some non real compia coefficients the feve command computes al real and complex roots It may met reum al roots for exceptenaly ill condpened polynemias For apama gaia of sytem of equations the feolve command computes a angle real root Y Examples Y A Polynomial Equation in One Variable Fee amave real petynemial equaien the twelve command competes all real olution piman sir jid 0 12 4006 gt ava polpecem aa 32490 35H1 0 SPITALIN iA Li Y Other Equations Fee mere complicated equations the salve command computes ene teal solution gt peal ids 3245 V7 ry sxy ly 1 gt ola polnemials ee SIUESOUORE ye OF 42 1 gt Foe tan gala 1 OP 423 oe ol ei os 1h Ao gt Figure 1 17 Example Help Page Every help page in Ma
173. e 334 Creating a Question Create documents for automated testing Viewing Questions in Maple and assessment Saving Test Content Worksheet Compatibility page 335 Classic Worksheet interface does not support all Standard Compatibility Issues Worksheet interface features 7 2 Document Formatting 283 7 2 Document Formatting To begin create a new Maple document From the File menu select New Document Mode For this example you can copy and paste text from any file The example text below is from a Maple help page plot but the formatting has been removed for demonstration purposes Copy and Paste You can cut copy and paste content within Maple documents and from other sources To copy an expression or part of an expression to another location on the document 1 Select the expression or part of the expression to copy 2 From the Edit menu select Copy 3 Place the cursor at the insertion point 4 From the Edit menu select Paste Result plot create a two dimensional plot Calling Sequence plot f ploti x x0 x1 plotivl 72 Parameters f expression in independent variable x independent variable X0 x1 lett and right endpomts of horizontal range Wl v2 z coordinates and y coordinates If you paste into a math input region Maple interprets all the pasted content as input If you paste into a text region Maple interprets all the pasted content as text However note tha
174. e a bookmark to designate a location in an active document This bookmark can then be accessed from other regions in your document or by using hyperlinks in other documents To display bookmark formatting icons activate the Marker feature From the View menu select Markers Oo gt Section 1 1 Bookmark Indicator Section 1 2 Figure 7 19 Bookmark Indicator 328 7 Creating Mathematical Documents Note You can display bookmark properties by holding the pointer over a bookmark indic ator See Figure 7 19 Inserting Renaming and Deleting a Bookmark To insert a bookmark 1 Place the cursor at the location at which to place the bookmark For example place the cursor in the Parameters section title 2 From the Format menu select Bookmarks The Bookmark dialog opens listing existing bookmarks in the document 3 Click New The Create Bookmark dialog opens See Figure 7 20 Enter a bookmark name parameters and click Create F Create Bookmark Figure 7 20 Create Bookmark Dialog 4 The new bookmark appears in the Bookmark dialog list Click OK 7 7 Embedded Components 329 Note You can also rename and delete bookmarks using the Bookmark dialog Result plot create a two dimensional plot Calling Sequence plottf x plottf x x0 x1 plot w1l v2 Parameters Bookmark parameters nression in independent variable z F independent variable 0 x1 let and night endpo
175. e are all of the solutions in the interval 0 7 Example 4 Find the Inverse Function If f x x 1 x 0 find and graph the rule for f x its functional inverse We solve this problem using the following methods e Implement the Definition Graphically page 226 Solution by Tutor page 229 Implement the Definition Graphically The graph of the inverse function is the set of ordered pairs formed by interchanging the ordinates and abscissas 1 In a blank document block enter Ea 1 x and press Enter 5 8 Clickable Math Examples 227 2 Right click the output and select Plots Plot Builder 3 In the Plot Builder Select Plot Type dia log ensure that 2 D parametric plot is selec ted in the Select Plot region 4 Adjust the domain for x to the interval 0 1 5 Click Plot to return the plot to the document 3 Interactive Plot Builder Select Plot Type E4 Select Plat Type and Functions Plot Select Plot 2 D parametric plot 2 D plot 2 D polar plot 3 D conformal plot of a complex valued Function 2 D conformal plot of a complex valued Function 2 D complex plot 3 D complex plot Edit Functions Select Variable Purposes Ranges and Plot Options ptions On Plot return plot command 228 e 5 Mathematical Problem Solving 6 Ctrl drag the expression x 1 onto this graph Notice that the axis ranges alter 7 Ctrl drag the expression onto this graph Th
176. e exits the loop The from by and to clauses are optional and can be in any order between the for clause and the do keyword Table 9 1 lists the default clause values Table 9 1 Default Clause Values infinity Examples The following loop returns the square root of the integers to 5 inclusive gt for n to 5 do evalf sqrt n end do 1 414213562 1 732050808 bh 2 236067977 When the value of the counter variable n 1s strictly greater than 5 Maple exits the loop gt n 6 The previous loop is equivalent to the following for from statement 9 2 Flow Control 375 gt for n from 1 by 1 to 5 do evalf sqrt n end do 1 414213562 732050808 2 236067977 The by value can be negative The loop repeats until the value of the counter variable is strictly less than the final value gt for n from 10 by 1 to 3 do if isprime n then print n end if end do 7 5 3 gt n 2 for in Loop The for in loop statement repeats a statement sequence for each component operand of an expression for example the elements of a list 376 9 Basic Programming Syntax The for in loop has the following syntax for variable in expression do statement sequence end do The for clause must appear first The behavior of the for in loop 1s 1 Assign the first operand of expression to the name variable 2 Execute the statement_sequence 3 Assign the next ope
177. e input and execution group boundaries Clear the Show input and Show ex ecution group boundaries check boxes Formatting Table Headers The following table uses cell merging for formatting row and column headers and row and column grouping to control the visibility of cell boundaries By default invisible cell boundaries are visible on mouse pointer roll over You can hide the visibility of lines on mouse pointer roll over by using the View Show Hide Contents dialog and clearing the Hidden Table Borders check box 7 4 Tables 317 Parameter 2 Low High Parameter Low 13 24 High 18 29 Table settings 1 Insert a table with 4 rows and 4 columns and enter the information shown above Using the Table menu 2 Merge the following sets of Row Column cells R1 C1 to R2 C2 R1 C3 to R1 C4 and R3 C1 to R4 C1 3 Group columns 1 and 2 and columns 3 and 4 4 Group rows and 2 and rows 3 and 4 In the Properties dialog Table Properties menu 5 Set Exterior Borders to None 6 Optional Change Table Size Mode size option to Scale with zoom factor Using the Table menu 7 Set Alignment of columns 3 and 4 to Center 2 D Math and Plots The following example illustrates the use of tables to display 2 D Math and plots side by side aot r f i Plotofe and its rational approximation Approximating exp x as a rational polynomial 0 8 using a 3 order Pad approximation 0 6 l 9 l 3
178. e resulting graph shows f x f x and the line y x 5 8 Clickable Math Examples 229 Adjust the x and yV axis ranges 8 Right click the plot and select Axes Properties 9 In the Axis Properties dialog de select Use data extents and change the range to 0 to 2 10 Click the Vertical tab and repeat step 9 Click OK to apply these settings and close the dialog Solution by Tutor 1 Load the Student Calculus 1 package Loading Student Calculus1 From the Tools menu select Load Package Student Calculus 1 2 Enter the expression x 1 inablank document block 3 Right click and select Tutors Calculus exis i Function inverse Single Variable Function Inverse The Function Inverse Tutor displays Enter a function and an interval Fix x 2 1 a 0 4 Adjust the domain to 0 2 Formula of the Inverse x 1 1 2 x 1 1 2 Plot Options Maple Command InverseP lot x 2 1 r Aa E PEA 230 e e 5 Mathematical Problem Solving 5 When you are finished click Close The A plot of the function its inverse and the line inverse tutor Y x is returned to the document Example 5 Methods of Integration Trig Substitution Evaluate the integral eS dy by making the substitution x 2 sin u 4 x We solve this problem using the following methods e Immediate Evaluation of the Integral page 230 Solution by Integration
179. e section heading 4 Press the Enter key 5 Enter the body of the section Tips for Adding Subsections The insert location of subsections is the same as for sections with a few exceptions e Subsections are inserted at the current cursor location when in a subsection e To insert a subsection immediately after the current subsection collapse the subsection and place the cursor in the subsection title Using the Indent and Outdent Toolbar Icons You can shift sections to create or remove subsections Enclose the selection in a section or subsection Outdent the selection to the next section level if possible For example to create two sections containing the two categories of information in the pasted text 1 Select Parameters and all of the items under it 2 Click the Indent toolbar item 3 Cut and paste Parameters from inside the section to its title 7 2 Document Formatting 297 4 Similarly create a section with the title Calling Sequence containing the items under that heading Result plot create a two dimensional plot Calling Sequence plottf x plottf x x0 x1 plot v1 v2 Parameters f expression in independent variable x independent variable 0 el left and right endpoints of horizontal range x 72 coordinates and y coordinates Note the section titles are automatically formatted as section titles but you can change the formatting thr
180. e set of right single quotes gt i54 368 8 Maple Expressions gt EY ial 8 11 gt 8 11 i 8 12 gt 8 12 5 8 13 For information on equation labels and equation label references see Equation La bels page 97 Enclosing an expression in unevaluation quotes delays evaluation but does not prevent automatic simplification gt ger sg 4q i 8 14 Unassigning a Name Using Unevaluation Quotes To unassign a name e Assign the name enclosed in unevaluation quotes to itself You can also unassign a name using the unassign command For more information see Unassigning Names page 95 9 Basic Programming You have used Maple interactively in the previous chapters sequentially performing oper ations such as executing a single command Because Maple has a complete programming language you can also use sophisticated programming constructs In Maple you can write programs called procedures and save them in modules These modules can be used and distributed in the same way as Maple packages Important It is strongly recommended that you use the Worksheet mode and 1 D Math input when programming or using programming commands Hence all input in this chapter is entered as 1 D Math 9 1 In This Chapter constructs Conditional Execution if Statement Repetition for Statement Iterative Commands page 380 Specialized Creating a sequence efficient iterative commands Adding and Multip
181. e sum of two fractions 3 2 Input Prompt 81 2 fi gt 9 1 85 99 3 2 Suppressing Output To suppress the output enter a colon at the end of the input a 2 gt 9 1 A set of Maple input and its output are referred to as an execution group 1 D Math Input You can also insert input using Text mode 1 D Math The input is entered as a one dimen sional sequence of characters 1 D Math input is red To enter input using 1 D Math e At the input prompt press F5 or click the Text button in the toolbar T switch from 2 D Math to 1 D Math gt 123 2 29857 120 1785623 120 Important 1 D Math input must end with a semicolon or colon If you use a semicolon Maple displays the output if you use a colon Maple suppresses the output gt 123 2 29857 120 To set the default input mode to 1 D Math 1 From the Tools menu select Options The Options dialog is displayed 2 On the Display tab in the Input display drop down list select Maple Notation 3 Click Apply to Session to set for only the current session or Apply Globally to set for all Maple sessions To convert 2 D Math input to 1 D Math input 82 3 Worksheet Mode 1 Select the 2 D Math input 2 From the Format menu select Convert To and then 1 D Math Input Important In Document mode you can execute a statement only if you enter it in Math mode Input Separators In 1 D and 2 D Math input you can use
182. efficients In the Matrix palette 1 Specify the size of the matrix for example 3 x3 2 In the Shapes drop down list select Diagonal 3 In the Data type drop down list select integer 1 4 Click the Insert Matrix button 5 Enter the values in the diagonal entries 5 3 Linear Algebra 165 2 3 0 0 gt 0 17 OF 0 O 32 You cannot specify properties when defining vectors using the angle bracket notation You must use the Vector constructor To define a column vector using the Vector constructor specify e The number of elements If you explicitly specify all element values this argument is not required e A list of expressions that define the element values e Parameters such as shape datatype and fill that set properties of the vector The following two calling sequences are equivalent gt Vector 0 0 0 0 0 gt Vector 3 shape zero 0 0 To create a row vector using the Vector constructor include row as an index gt Vector row 3Jfill 1 111 gt Vector row 127 0 34 datatype integer 1 127 0 34 The Matrix palette does not support some properties To set all properties use the Matrix constructor 166 5 Mathematical Problem Solving To define a matrix using the Matrix constructor specify e The number of rows and columns If you explicitly specify all element values these ar guments are not required e A list of lists that define the
183. ensional limits refer to the limit multi help page Numerically Computing a Limit To numerically compute a limit e Use the evalf Limit arguments calling sequence Important Use the inert Limit command not the limit command For more information refer to the limit help page The Limit command accepts the same arguments as the limit command For example gt evalf Limit a l 225 cos x tan x Q 3020605357 For information on the evalf command see Numerical Approximation page 361 The Limit command does not compute the limit It returns an unevaluated limit 5 4 Calculus 177 sin x gt LMU n r me l cos x tan x x 1 225 him sin x E ae te ae ERN x 1 225 cos x tan x For more information on the Limit command refer to the Limit help page Differentiation Maple can perform symbolic and numeric differentiation To differentiate an expression d 1 In the Expression palette click the differentiation item dx i or the partial differentiation item ar 2 Specify the expression and independent variable and then evaluate it For example to differentiate xsin ax with respect to x gt xsinfaa dx sin ax xcos ax a You can also differentiate using context menus For more information see Context Menus page 39 To calculate a higher order or partial derivative edit the derivative symbol inserted For example to calculate the second derivati
184. enter mathematical content that is evaluated enter it at an Input Prompt page 80 e Insert any palette item Palette items are inserted in Math mode 2 D Math Note After you insert a palette item you must press F5 or click the toolbar icon to return to Text mode You can format text in a text region Features include e Character styles e Paragraph styles e Sections and subsections e Tables For more information on formatting documents see Creating Mathematical Docu ments page 281 94 3 Worksheet Mode 3 9 Names Instead of re entering an expression every time you need it you can assign it to a name or add an equation label to it Then you can quickly refer to the expression using the name or an equation label reference For information on labels see the following section Equation Labels page 97 Assigning to Names You can assign any Maple expression to a name numeric values data structures procedures a type of Maple program and other Maple objects Initially the value of a name 1s itself gt a a The assignment operator associates an expression with a name gt ai t d Recall that you can enter 72 using the following two methods e Use the Common Symbols palette In 2 D Math enter pi and then press the symbol completion shortcut key See Shortcuts for Entering Mathematical Expressions page 7 When Maple evaluates an expression that contains a name it replaces the name with
185. ents subpackage contains the elements available when designing a Maplet application After you define the Maplet use the Maplets Display command to launch the Maplet For more information on the Maplets package refer to the MapletsPackage help page For more examples of designing Maplets using the Maplets package see the Maplets Roadmap help page Example 4 Design a Maplet Using the Maplets Package To introduce the structure of designing Maplets using the Maplets package this example illustrates the equivalent syntax for the Design a Maplet Using the Maplet Builder page 404 408 10 Embedded Components and Maplets Load the Maplets Elements package gt with Maplets Elements Define the Maplet application To suppress the display of the data structure associated with the Maplet application end the definition with a colon gt PlottingMaplet Maplet BoxLayout BoxColumn First Box Row BoxRow Define a Plot region Plotter reference Plotterl1 End of first Box Row Second Box Row BoxRow Define a Label Label Enter a function of x Define a Text Field TextField reference TextFieldl Define a Button Button caption Plot Evaluate value plot TextFieldl x 10 10 target Plotterl1 End of second Box Row End of BoxColumn End of BoxLayout End of Maplet Launch the Maplet gt Maplets Display PlottingMaplet
186. er Operations 109 gt convert 6000 binary 1011101110000 gt convert 34271 hex SIDF For information on enclosing keywords in right single quotes see Delaying Evalu ation page 366 You can also use the convert base command gt convert 34271 base 16 15 13 5 8 Note The convert base command returns a list of digit values 1n order of increasing signi ficance Finite Rings and Fields Maple supports computations over the integers modulo m The mod operator evaluates an expression over the integers modulo m gt 27 mod 4 By default the mod operator uses positive representation modp command Symmetric representation is available using the mods command gt modp 27 4 gt mods 27 4 For information on setting symmetric representation as the default refer to the mod help page The modular arithmetic operators are listed in Table 4 2 110 4 Basic Computations Table 4 2 Modular Arithmetic Operators Addition T gt 7 6mod5 o Subtraction gt mods 3 16 11 Multiplication displays in 2 D Math as 13 5 mod 3 Multiplicative inverse displays in 2 D Math as a superscript Division displays in 2 D Math as Exponentiation 100 amp 100 mod 7 2 To enter a caret in 2 D Math enter a backslash character followed by a caret that is For information on solving an equation modulo an integer see Integer Equations in a Finite Field p
187. er is just a variable and is not the same as the imaginary unit y 1 denoted by I or i in Maple Multiply two complex numbers 0 123 0 7451 and 4 2 1 Action Result in Document 1 In anew document block enter 0 123 0 745 il 0 123 0 745 i 2 Press the symbol completion shortcut 0 123 0 745 i key Esc Maple displays a pop up list of partial and exact matches including symbols and commands loomfent icontent icosahedron location nloffools icosahedron M x p 27 icosahedron location scale ploffeols icosahedron i x p 217 ve E tla 3 Select the imaginary unit 0 123 0 745 i imaginary 1 F 4 Close the parentheses enter a space 0 123 0 7451 4 2 i for implicit multiplication and type the second expression in parentheses using symbol completion for the second ima ginary number 5 Press Ctrl Command 0 123 0 745i 4 2 i 0 2284 3 25201 Macintosh to evaluate the product 1 2 Entering Expressions 31 For more information on entering complex numbers refer to the 7HowDol help page Toolbar Icons In the introduction section you learned about the toolbar icons and context toolbars available in Maple see Toolbar Options page 9 The toolbar can be used to format your document alter plots and animations draw in a canvas write in both Math and Text modes in one line and much more The last of these is demonstrated in the next exa
188. erics SSCS mal Compute numeric product SOS select select Return operands that satisfy a condition Return operands that do not satisfy a condition selectremove Return operands that satisfy a condition and separately return operands that do not satisfy a condition map si Apply command to the operands of an expression Apply binary command to the operands of two lists or vectors Creating a Sequence The seq command creates a sequence of values by evaluating a specified expression over a range of index values or the operands of an expression See Table 9 3 Table 9 3 The seq Command seq expression name initial final gt seq exp x x 2 0 seq expression name in expression gt seq u u in Pi 4 Pi 2 2 1 Pi E foe S 4 gt a 9 3 Iterative Commands 381 Adding and Multiplying Expressions The add and mul commands add and multiply sequences of expressions over a range of index values or the operands of an expression See Table 9 4 Table 9 4 The add and mul Commands add expression name initial final gt add exp x x 2 4 mul expression name initial final e 4 e a et gt mul 2 x x 1 10 3715891200 add expression name in expression gt add u u in Pi 4 Pi 2 Pil mul expression name in expression gt mul u u in Pi 4 Pi 2 Pi A a ar 8 The endpoints of the index range initial and final in the add and mul calling sequence
189. ess the roots in eval 16 floating point form 0 8480d20790 2 299530575 3 481429564 17 5943348398 3 Replace the current equation with the one Solve Analytically in a Specified Interval from this example on i ple Enter an expression a sas se amp E F 6cos x cos x 2 0 and then 6 cos x cos x 2 0 Find th ts 1 execute the commands Notice that equation 22n nema gt Studenti Calculus Roots 15 0 2 7 specified interval 24 2 4 2 labels are used to reference the results arceos y T arccos 16 2 Express the roots in epalf 16 floating point form 0 8410686708 2 094395103 4 188790204 17 5442116637 Analytic Solution 1 Ctrl drag the equation 6 cos x cos x 2 0 6 cos x cos x 2 0 toa blank docu ment block region 7 a the expression and select Left hand 6 cog Was lt 2 0 left hand side y 226 5 Mathematical Problem Solving 3 Right click the output and select Factor 6 cos x cos x 2 coslx 1 3cos x 4 Ctrl drag the first factor to a blank document block region 2 cos x 1 5 Right click and select Solve Solve 6 Ctrl drag the second factor to a blank docu ment block region 3 cos x 2 7 Right click and select Solve Solve Notice that you have not found all of the solutions as with the above methods Thes
190. essel Mathieu Heun and Legendre families of functions Units Calculator an interface to convert between 500 units of measurement Worksheet Migration an interface to convert worksheets from Classic Maple mws files to Standard Maple mw files CAD Link an interface to explore the properties of models from supported CAD applic ations available on Microsoft Windows only Tutors Maple provides over 40 interactive tutors to aid in the learning of Precalculus Calculus Multivariate Calculus 38 1 Getting Started e Vector Calculus Differential Equations e Linear Algebra e Complex Variables These tutors are easily accessible in the Tools menu by selecting Tutors See Figure 1 6 Tools Window Help Assistants eo ao ol oc rh Tutors b Calculus Multi Variable gt Tasks b Calculus Single Variable gt Complex Variables e Load Package l l l Er ae Differential Equations Unload Package I Linear Algebra b Eigenvector Plot Spellcheck F7 Numerical Analysis Eigenvalues Complete Command Precalculus b Eigenvectors Help Database b Vector Calculus b Gauss Jordan Elimination Gaussian Elimination Options Linear System Flot Check For Updates Linear Transform Plot Matrix Builder Matrix Inverse Linear System Solving Figure 1 6 Accessing Tutors from the Tools Menu Some of the tutors can also be accessed through the Student package The Differenti
191. essible As such users students can switch between the original document contents and the displayed assignment 7 10 Worksheet Compatibility Maple provides users with two worksheet interfaces the Standard Worksheet and the Classic Worksheet Both have access to the full mathematical engine of Maple and take advantage of the new functionality in Maple The Classic Worksheet has the traditional Maple worksheet look and uses less memory If you create a document in the Standard Worksheet interface of Maple and then open it in the Classic Worksheet interface you should note possible changes to your file For example a bulleted list in the Standard Worksheet will not be displayed with bullets in the Classic Worksheet Many of the graphical features in this manual especially those in this chapter are not available in the Classic Worksheet interface If you are creating documents for distribution refer to the Compatibility help page 336 7 Creating Mathematical Documents 8 Maple Expressions This chapter provides basic information on using Maple expressions including an overview of the basic data structures Many of the commands described in this chapter are useful for programming For information on additional Maple programming concepts such as looping conditional execution and procedures see Basic Programming page 369 8 1 In This Chapter Creating and Using Data Structures page 337 Expression Sequences How to define
192. ession using the units palettes to insert units 2 Right click Control click for Macintosh the expression 3 From the context menu select Units and then Simplify For example compute the electric current passing through a wire that conducts 590 coulombs in 2 9 seconds 590 C simplify units 203 4482759 A 291s lA 76 2 Document Mode For more information on using units see Units page 128 Assistants and Tutors Assistants and tutors provide point and click interfaces with buttons text input regions and sliders For details on assistants and tutors see Point and Click Interaction page 32 Assistants and tutors can be launched from the Tools menu or the context menu for an ex pression For example you can use the Linear System Solving tutor to solve a linear system specified by a matrix or a set of equations Example 3 Using a Context Menu to Open the Linear System Solving Tutor Use the Linear System Solving tutor to solve the following system of linear equations written in matrix form Ll 2 Q Z 4 4 2 1 5 7 Mos 53 4 Sf l 1 3 6 5 1 In a new document block create the matrix or set of linear equations to be solved 2 Load the Student LinearAlgebra Loading Student LinearAlgebra package From the Tools menu select Load Package Student Linear Al gebra This makes the tutors in that package available For details see Pack age Commands page 47 2 6 Performing Computations
193. ette Categories Palette Category Palette Description Expression Palettes b fre OW Matrix Expression construct expressions such as integrals a Rows Matrix enter the number of rows and columns required designate Columns type such as zero filled and designate shape such as diagonal Choose i i p Ehon Layout add math content that has specific layout such as expressions Type Custom values Y Shape Any with one or more superscripts and subscripts Data type Any I Components embed graphical interface components such as a button ee into your document or worksheet Components can be programmed to perform an action when selected such as executing a command when a button is clicked Toggle Button Handwriting an easy way to find a desired symbol Units SI insert a unit from the International System of Units SI or any general unit te Units FPS insert a unit from the Foot Pound Second System FPS or any general unit L7 f Accents insert decorated names such as an X with an arrow over it to denote a vector 4 Favorites add templates that you use most often from other palettes 24 e Getting Started Palette Category Palette Description Mathematical Palettes Palettes for constructing expressions JF common Symbols Common Symbols Relational Relational Round Operators i Large Operators i Negated F Fenced Arrows gt
194. f Units A dimension is a measurable quantity for example length or force The set of dimensions that are fundamental and independent are known as base dimensions In Maple the base dimensions include length mass time electric current thermodynamic temperature amount of substance luminous intensity information and currency For a complete list enter and execute Units GetDimensions Complex dimensions or composite dimensions measure other quantities in terms of a combination of base dimensions For example the complex dimension force is a measurement mass length of time Each dimension base or complex has associated units Base units measure a base dimension Complex units measure a complex dimension Maple supports over 40 units of length in cluding feet miles meters angstroms microns and astronomical units A length must be measured in terms of a unit for example a length of 2 parsecs Table 4 4 lists some dimensions their corresponding base dimensions and example units 4 5 Units Scientific Constants and Uncertainty 129 Table 4 4 a Dimensions Diei _ time second minute hour day week month year millennium blink lune lenothe mass joule electron volt erg watt hour APE OMASE calorie Calorie British thermal unit time Electric potential volt abvolt statvolt length mass eo time electric current For the complete list of units and their contexts and symbols
195. f x a lt x lt b about an axis or a line parallel to an axis 5 7 Teaching and Learning with Maple 207 ___ Tutors Check for Existing Tools ____ Task Templates __ Help Pages Check for Instructions Example Worksheets _ Applications X Check for it a Other Ready Teu Maple to Use Resources Application Center Figure 5 14 Flowchart of solving a problem Check for Existing Tools Tutor Begin by examining the Tools menu for a Tutor to a Volume of Revolution problem To access a Tutor for the Volume of Revolution 1 From the Tools menu select Tutors and then Calculus Single Variable Notice that a Volume of Revolution tutor exists 208 5 Mathematical Problem Solving 2 Click the Volume of Revolution menu item The following Maple command is entered in your document gt Student Calculus VolumeOfRevolutionTutor _ 5 7 Teaching and Learning with Maple 209 The Volume of Revolution Tutor is displayed See Figure 5 15 Use this tutor to enter a function and an interval view and manipulate the corresponding plot and view the full Maple command associated with your entries and selections Calculus 1 Volume of Revolution Plot window Enter 1 or 2 Functions and an interwal Fix 1 10 cosf1O sx gls OOo a 0 Method midpoint wt Number of partitions 6 Riemann sum Volume of the Solid 6
196. few options that are not available for static plots Refer to the animate help page for information on these additional options By default a two di mensional animation consists of sixteen plots frames and a three dimensional animation consists of eight plots frames To create a smoother animation increase the number of frames using the frames option Note Computing more frames increases time and memory requirements ZE gt sinewave Pe sats g x 0 m gt ball proc x y plots pointplot x y symbol circle symbolsize 20 end proc 278 6 Plots and Animations j t sin t e gt Pe 20 frames 60 background sinewave gt plots animate m f 6 8 Exporting 279 6 8 Exporting You can export a generated plot or animation to an image in various file formats including DXF and X3D for 3 D plots EPS GIF JPEG JPG POV Windows BMP and WME Exporting an animation to GIF produces an animated image file The exported images can be included in presentations web pages Microsoft Word or other software To export an image 1 Right click the plot region Control click for Macintosh 2 Select Export and the file format Alternatively 1 Click the plot 2 From the Plot menu select Export and then the file format Maple has various plot drivers By setting the plotdevice a file can be automatically created without returning the image to the document For more information refer
197. first construct the skeletal structure that is foundation floors and walls and then proceed to add the windows and doors Constructing a Maplet is no different First define the rows and columns of the Maplet application and then proceed to add the body elements such as buttons text fields and plot regions Simple Maplet A Maplet application can be defined using the commands in the Maplets Elements package and then launched using the Maplets Display command The following commands define and run a very simple Maplet application that contains the text string Hello World gt with Maplets Elements gt MySimpleMaplet Maplet Hello World gt Maplets Display MySimpleMaplet ry Maplet Hello World Figure 10 6 A Simple Maplet 10 5 Authoring Maplets 403 Maplet Builder To start the Maplet Builder e From the Tools menu select Assistants Maplet Builder Layout Pane Maplet Builder Untitled Maplet File Help Jal Button background ie z y caption Elle e A enabled true s ae font foreground Y Dialog image E z Z z Z onclick clickButton1 E reference Buttoni oottip TMe Menu visible true a f X ToolBar Di Y Other AE RIAA Aj a J Layout 4 Ton 4 af at Action Runvvindow c
198. for students math educators and engineers The Maple Portal includes e How Do I topics that give quick answers to essential questions e Tutorials that provide an overview of topics from getting started to plotting data manip ulation and interactive application development e Navigation to portals with specialized information for students math educators and en gineers Access the portal from the Help menu Help Manuals Resources and More Maple Portal Applications and Example Worksheets Applications Sample applications demonstrate how Maple can be used to find and document a solution to a specific problem Some applications allow for input or contain animations that you can run however their primary use is for demonstrations Topics include DC Motor Control Design Digital Filter Design Frequency Domain System Identification Harmonic Oscillator Image Processing and Radiator Design with CAD Systems Examples 1 6 Available Resources 59 Example worksheets are executable documents covering topics that demonstrate syntax or invoke a user interface to make complex problems easy to solve and visualize You can copy and modify the examples as needed Topics include Algebra Calculus Connectivity Discrete Mathematics General Numerics and Symbolics and Integral Transforms e From the Help menu select Manuals Resources and more and then Applications and Examples Manuals You can access all of Maple s ma
199. g Matrices and VECIOLS cucenschtscecceagnaninesesieersewieretsaseeinsanyneeieaeeeess 159 Accessing Entries in Matrices and Vectors cccceceseeceecenseeeeeesenseneenes 166 Linear Algebra Computations 4 c nexesearnceccenesentvetacssaie caste eeeterierenetauaee enna 168 Student LinearAlgeebra Package sorrsirerssicrorrssisorresisstreitsnt n idan EENE ENAN 174 CCU a EE E EE E co ag E T 175 LO r A A EA EE E EE E 175 D AO r E E EE EEEE E EEEE 177 SOOD a E E T E E E E 182 AVC eE o E A E SE EE A A E E EE 183 Dillerental Faua MOUS sing stresandaesadarancrenaeekcesdsemieneneweinnd samt numneteentegarenneriied 186 ACUI PACA GS an arenas ase ven ura atumaaane sees E A 186 So UIA OU sarrera E E EEEE E EEE EEEE EEEE 188 P intand C iCk Interit sg cccnmc cease meteermdeates nce ie E EEE EE AEREE 188 Large Optimization Problems tp chix evedosiece hearers mcseceasorinnetocuiseaonaeepeerenrs 191 MPS X File UD ORE seperi erri rionn EEES CENERE NENEN EEEE EEEE EEA 192 Optimization Package Commands scarscnanccindsasiesescctesseriereusddenrsaaxrevvensierenes 192 DO OS ee EEE ESE EAEE E E EEE 193 Probability Distributions and Random Variables ccc cceccec eee eneeeeeeees 193 Bytes CICA COMING sorrir anren neces ea eens eaten ae epee eea eee cues 194 PUTIN ore ee ae ec E i ee en ep ee A ee 195 Additional Information 4 nantacieytecnsereeeapecnsedeeavatarirabearksnssaceadsascneinenetien 198 5 7 Teaching and Learning with Maple ssdcecsvswriesovdoc
200. g the assistant from the context menu that displays 34 Getting Started Tools Window Help Assistants k Back Solver Tutors b Curve Fitting Tasks b Data Analysis Equation Manipulator Load Package b Import Data ANAE eee Installer Builder Spellcheck F7 Library Browser Complete Command Maplet Builder Help Database ODE Analyzer l Optimization aad Plot Builder Check For Updates Scientific Constants Special Functions Units Calculator Worksheet Migration CAD Link Figure 1 5 Accessing the Assistants from the Tools Menu Example 7 Curve Fitting Assistant Enter a data sample and use the Curve Fitting Assistant to find the best approximation of a function to fit the data Result in Document 1 From the Tools F menu select Assistants Curve Fitting Assistant Curve Fitting The Enter data points below first dialo g inthe Independent Values x Dependent Values F x Curve Fitting Assist ant appears Current Page 1 Previous 1 3 Point and Click Interaction 35 Acton Result in Document 2 Enter data as Inde ra pendent Values and Curve Fitting Assistant Dependent Values Enter data points below Alternatively you Independent Values x Dependent Values F x could import a file con Baar taining data If you 0 044194 have more data than the o space provided click 0 044194 the Next Page button 0 176777 for more spac
201. ge 137 136 4 Basic Computations Element Properties Maple also contains element properties and isotope properties Elements Maple supports all 117 elements of the periodic table Each element has a unique name atomic number and chemical symbol You can specify an element using any of these labels For a complete list of supported elements refer to the ScientificConstants elements help page Maple supports key element properties including atomic weight atomicweight electron affinity electronaffinity and density For a complete list of element properties refer to the ScientificConstants properties help page Isotopes Isotopes variant forms of an element that contain the same number of protons but a different number of neutrons exist for many elements To see the list of supported isotopes for an element use the GetIsotopes command gt Getlsotopes element L1 Li Li Lig Li Li Lig Li 9 Li Lija Maple supports isotopes and has a distinct set of properties for isotopes including abundance binding energy bindingenergy and mass excess massexcess For a complete list of isotope properties refer to the ScientificConstants properties help page Accessing an Element or Isotope Property Definition The GetElement command in the ScientificConstants package returns the complete definition of an element or isotope 4 5 Units Scientific Constants and Uncertainty 137 gt GetElemen
202. ges Display a Parametric Plot Some graphs cannot be specified explicitly In other words you cannot write the dependent variable as a function of the independent variable y x One solution is to make both the x coordinate and the y coordinate depend upon a parameter 256 6 Plots and Animations gt plot cos 31t sin 5 t 0 27 Display a 3 D Plot Maple can plot an expression of two variables as a surface in three dimensional space To customize the plot include plot3d options in the calling sequence For a list of plot options see The plot and plot3d Options page 267 6 2 Creating Plots 257 gt plot3d ae py 2 v 2 2 glossiness 0 5 style patchnogrid light 100 345 0 4 0 9 0 7 ambientlight 0 5 0 1 j The plots Package The plots package contains numerous plot commands for specialized plotting This package includes animate contourplot densityplot fieldplot odeplot matrixplot spacecurve textplot tubeplot and more For details about this package refer to the plots help page gt with plots The pointplot Command To plot numeric data use the pointplot command in the plots package with the data organ ized in a list of lists structure of the form Er A x V gt e xp Yall By default Maple does not connect the points To draw a line through the points use the style line option For further analysis of data points use the Curve Fitting Assistant Tools
203. ght endpomts of horizontal range wl 2 z coordinates and y coordinates Attributes Submenu Setting Fonts Character Size and Attributes You can also change various character attributes such as font character size style and color in one dialog To modify text 1 In the document select text to modify 2 From the Format menu select Character and then Attributes The Character Style dialog opens See Figure 7 2 286 7 Creating Mathematical Documents E Character Style E Underlined Superscript Angsana Mew Subscript AngsanalPe Arabic Transparent Figure 7 2 Character Style Dialog Quick Paragraph Formatting The Format Paragraph menu provides access to the following quick alignment features Align Left Center Align Right and Justify To modify a paragraph 1 In the document select the paragraph to modify 2 From the Format menu select Paragraph and then the appropriate feature Attributes Submenu Spacing Indent Alignment Bullets Line Break and Page Break You can change various paragraph attributes in one dialog e From the Format menu select Paragraph and then Attributes The Paragraph Style dialog opens See Figure 7 3 7 2 Document Formatting 287 e When changing spacing you must indicate units inches centimeters or points in the Units drop down list Pa ragraph Style Properties Units pt Spacing Indent Line 0 0 Left Margin 0 0 Above
204. h ScientificErrorAnalysis gt Quantity 3 5 m 0 1 m ScientificErrorAnalysis Ouantity 3 5 m 0 1 m For a relative error you can specify the units in only the value gt Quantity 3 5 m 0 1 relative ScientificErrorAnalysis Quantity 3 5 m 0 35 m For information on the correlation between variance of and covariance between quantities with uncertainty refer to the ScientificErrorAnalysis help page Performing Computations with Quantities with Uncertainty Many Maple commands support quantities with uncertainty gt gl QOuantity 31 2 gt g2 QOuantity 20 1 Compute the value of the derivative of g x sin q2 x at x sin 7 4 142 4 Basic Computations gt dl diff ql x sin q2 x x di 2 ScientificErrorAnalysis Quantity 31 2 x cos ScientificErrorAnalysis Ouantity 20 x ScientificErrorAnalysis Quantity 20 1 gt d2 eval dt x sin To convert the solution to a single quantity with uncertainty use the combine errors command gt result combine d2 errors The value of the result 1s gt evalf result 43 74124725 The uncertainty of the result 1s gt GetError result 14 4269061 2 Additional Information For information on topics including e Creating new rounding rules e Setting the default rounding rule and e Creating a new interface to quantities with uncertainty refer to the ScientificErrorAnalysis help page
205. he Interactive Plot Builder the context menus or as options in the command syntax Interactive Plot Builder Options The Interactive Plot Builder offers most of the plot options available in Maple in an easy to use interface Example Launch the Interactive Plot Builder and enter the expression 1 Add the expression 2 x 5 10 x 3 6 x 1 For information on interacting with the Interactive Plot Builder see Example I Display a plot of a single variable expression page 240 Set the x axis range 2 In the Select Plot Type window change the x axis range to 2 2 In the Plot Options window 3 From the Line group box select dot from the left drop down menu 4 From the Color group box select Blue 5 From the Axes group box select frame 6 From the Title group box enter My Plot in the text field Plot the expression 7 Click Plot Context Menu Options Using the context menu you can alter a plot by right clicking Control click for Macintosh the plot output You can also access a large subset of plot options using the Plot toolbar and Plot menu options These menus display when a plot region 1s selected Regardless of the method used to insert a plot into Maple you can use the context menu to apply different plot options For a list of options available when plotting in two and three dimensions see The plot and plot3d Options page 267 2 D Plot Options Some plots do not display as you would expect using defau
206. he document context bar you can apply e Existing Maple paragraph styles e New styles that you have created through the Style Management Figure 7 4 and Paragraph Style Figure 7 6 dialogs To apply a Maple paragraph style to text in your document 1 Select the text to modify 2 In the styles drop down list in the context bar of your document select an appropriate paragraph style All Maple paragraph styles are preceded by the letter P The selected text now reflects the attributes of the paragraph style you have chosen P Annotation Title P Author P Bullet Iter P Dash Item P Diagnostic P Error P Fixed width P Heading 1 ae 7 2 Document Formatting 293 For example to format the title of the pasted text as a title first select the line plot create a two dimensional plot In the Styles drop down select Title Result plot create a two dimensional plot Calling Sequence plottf x plottf x x0 x1 plotivl v2 Parameters f expression in independent variable x F independent variable 0 xl left and right endpoints of horizontal range i 72 x coordinates and y coordinates 3 Optional If necessary you can remove this style From the Edit menu select Undo Creating and Modifying Paragraph Styles You can create custom paragraph styles to apply to text or change existing paragraph styles New styles are automatically added to the styles drop down list in the contex
207. he plot component and the function should display in the math container 5 Troubleshooting The first Do command gives an error because the second parameter is 0 One way to avoid this problem 1s to change the range of the second dial In the Component Properties dialog for the second DialComponent change the Value at Lowest Position from 0 to 1 Altern atively you could change the code to compensate with an if statement 6 Copy the actions to the components Once the commands work as expected you can copy them into the components e Open the Component Properties dialog for the first DialComponent and click the Edit button for Action When Value Changes Copy and paste the commands into the space between the use statements 10 3 Creating Embedded Components 399 Action When Value Changes use DocumentTools in parameteril Do sDiald parameterz Do tDiall Do RotaryGauge0 parameteri parameter Do Ploti plot parameterz x parameterl1 x S0 50 y S0 50 Do MathContainerO y parameter2 x parameterl Check syntax before saving Check Now Figure 10 5 DialComponent Action dialog e Do the same for the second DialComponent 7 Create the layout for the components Create a table and then cut and paste the components into it along with explanatory text Important you must cut not copy the components or their names will be changed to 400 10 Embedded Components and Maplets avo
208. he property of their respective owners This document was produced using a special version of Maple and DocBook Printed in Canada ISBN 978 1 8973 10 89 2 Contents PTAC acest pene E u ease nee E E EE E E tes eenesseeues xiii EE AU e E E A A T A E E nests l L Trod Hon 10 Maple sc2cs ca caacccenenesee assesaconsesevareqen sce aor coreeosanniseantncsaeaecs 2 Work nmaa Maple se crccucescececereecntevsoeausaecs eee venusgucepdecessaeeageetueeeerdscdceceurest 2 Starting the Standard Document Interface sc cssincnscrssccmcoancsaneateaavendasetentetanes det 3 F tening 2D Ma en ne nee ae eee 5 ToD 010010 eee te ee EE ne cee en eee eee ee 9 Context Men s and Copy amp Drap cag tiv ncentietstectecesietous eet ea EER etre eee 12 Saving a Maple Document si jcaqnncanaroseresdoncedunaeiuadoweseereenieees coreiuamrercterwann 19 1 2 Entering m4 0 io 0 8 ee ee eritre s oe ae ene eee ee 19 BCC UUIOM Groups a ccesene tgeoceeceesevannavewieeecteunaeseect qoveseenestete corm esetecoute eceetests 19 Mati Modo vs TXL MOQO geicancgtencuscancaceserannseneaseoucentueaiemeeneicae neaenstanasaaess 20 PoE E E A gen cea eee sereaesien ss 22 OGL NIOS sees crete eet eens ee enc es ne pawn nea ae eee eneeee nee 29 TOODA ICON eae E T E E E ee 31 1 3 Point and Click Interaction 4 pecneiensc onset essantcoarenaies eisai eeancsensennaencteneemesesd 32 ASSI e E EE AEE E EE E AEE E E EEE OANE AS EE EENE EE EEEO S 32 TOO eonen Enr E cee EERE E E EREE EE EES 37 Contei MOMS
209. hecking of the solution ScientificErrorAnalysis The Scientific Error Analysis package provides representation and con struction of numerical quantities that have a central value and an associated uncertainty or error which is a measure of the degree of precision to which the quantity s value is known Various first order calculations of error analysis can be performed with these quantities Statistics The Statistics package is a collection of tools for mathematical statistics and data analysis The package supports a wide range of common statist ical tasks such as quantitative and graphical data analysis simulation and curve fitting Package Name Student VectorCalculus 3 4 Palettes 3 4 Palettes 87 The Student package is a collection of subpackages designed to assist with teaching and learning standard undergraduate mathematics The many commands display functions computations and theorems in various ways including stepping through important computations The Student package contains the following subpackages Calculus single variable calculus LinearAlgebra linear algebra MultivariateCalculus multivariate calculus NumericalAnalysis numerical analysis Precalculus precalculus VectorCalculus multivariate vector calculus The Units package contains commands for unit conversion and provides environments for performing calculations with units It accepts approxim ately 300 distinct unit names for e
210. ht click the palette dock Maple displays show Palette Favorites Maple Cloud a context menu near the palette show All Palettes ae Show Default Palettes Expression atric 2 From the context menu select Show Expand all Palettes Matri Palette and then select the palette i alae Ai Petes Layout Components Expand Docks Handwriting Collapse Docks Units 51 Units FPS Arrange Palettes Accents Roman Extended Upper Case Roman Extended Lower Case Diacritical Marks e infa Greek lal log 4 log a Cyrillic Script sin a cos a tan a Open Face p fla fae Fraktur Common Symbols fray Relational f ahb oz Relational Round Operators xX xa Large Operators Negated Fenced ArFOWS Constants and Symbols Punctuation Miscellaneous To expand or collapse a palette in the palette dock Click the triangle at the left of the palette title F Expression 1 2 Entering Expressions 27 To move a palette in the palette dock Y Favorites Move the palette by clicking the title and dragging the palette to the new location b gt Expression pPeSonmon Symbols To expand or collapse the palette docks bee 55 XD b g U 3 Select the appropriate triangle at the top right G or top left side of the palette region t W Favorites F Expression k Example 3 Enter an Expression Using Palettes Review the following example 10 S 72 5i 2420 i 10 I
211. ication for which the Maple code is contained in a Maple document you need to execute the Maplet code To display the Maplet application you must use the Maplets Display command Note The Maplet code may be quite large if the Maplet ap plication is complex In this case execute the document to ensure user defined procedures that are referenced in the Maplet application are also defined Typical procedure 1 If present evaluate user defined procedures Myproc proc 2 Load the Maplets Elements package with Maplets Elements 3 Evaluate the Maplet definition Maplet_name Maplet Maplet definition 4 Display the Maplet application Maplets Display Maplet_name 402 10 Embedded Components and Maplets Important When a Maplet application is running you cannot interact with the Maple document 10 5 Authoring Maplets To author Maplets you can use the Maplet Builder GUI based or the Maplets package syntax based The Maplet Builder allows you to drag and drop buttons sliders text re gions and other elements to define the Maplet application and set the element properties to perform an action upon selection or update of the element The Maplet Builder is designed to create simple Maplets The Maplets package offers more capabilities control and options when designing complicated Maplet applications Designing a Maplet application is similar to constructing a house When building a house you
212. ications that help instructors begin using Maple and use Maple in the classroom Browse the many resources in the Education and Education PowerTools categories http www maplesoft com applications 5 7 Teaching and Learning with Maple 199 Resource Description Student Help Center The Maple Student Help Center contains tutorials and applica tions that help students learn how to use Maple explore math ematical concepts and solve problems Available resources in clude e Study guides Complete lessons with examples for academic courses including precalculus and calculus For example the Interactive Precalculus Study Guide contains worked prob lems each solved as in a standard textbook using Maple commands and custom Maplet graphical interfaces Free course lessons for many subjects including precalculus to vector calculus high school abstract and linear algebra engineering physics differential equations cryptography and classical mechanics Applications for students written by students providing ex amples in many subject areas e Student FAQs with answers from experts http www maplesoft com academic students Teacher Resource Center The Maple Teacher Resource Center contains resources and tips for teachers using Maplesoft products to help in the classroom Available resources include e Training videos e E books http www maplesoft com teachercenter Student Packages and Tutors
213. icons the icon that is selected remains in that state until prompted otherwise therefore if the Text icon is selected and you press the Enter key the new input region remains a Text region 12 e 1 Getting Started The Text and Math icons differ while at a Maple input prompt The Math icon displays input as 2 D Math whereas the Text icon displays Maple input For details refer to Math Mode vs Text Mode page 20 gt fi gt lee BY A To access the tools available in the Plot and Drawing icons click a plot region These tools allow you to manipulate the plot or draw shapes and enter text on the plot region By clicking an animation region you have the same features available for a plot region in ad dition to tools for playing the animation in the Animation icon For details on plots and animations refer to Plots and Animations page 237 For the remaining icons hover the mouse over the icon to display the icon description Context Menus and Copy amp Drag Context Menus Maple dynamically generates a context menu of applicable options when you right click an object expression or region The options available in the context menu depend on the selected input region For example you can manipulate and graph expressions enhance plots format text manage palettes structure tables and more When using context menus to perform an action on an expression the input and output are connected with a self docu menting arro
214. id duplication For information on creating and modifying tables refer to Tables page 306 Parameters a and Plot Window Use the Duals to set parameters tf 2 Result ry Plot of y 7x 89 10 4 Using Maplets A Maplet is a pop up graphical user interface that provides interactive access to the Maple engine through buttons text regions slider bars and other visual interfaces You can create your own Maplets and you can take advantage of the built in Maplets that cover numerous academic and specialized topics Built in Maplets include some assistants and tutors such as the ODE Analyzer For more information on this assistant see Ordinary Differential Equations ODEs page 121 Maplet applications are launched by executing Maplet code Maplet code can be saved in a Maplet maplet file or Maple document mw Maplet File To launch a Maplet application saved as a Maplet file e In Windows double click the file from a Windows file browser e In UNIX and on Macintosh use the command line interface At the command line enter maple q lt maplet_filename gt 10 4 Using Maplets 401 To view and edit the Maplet code contained within the maplet file 1 Start Maple 2 From the File menu select Open Maple displays the Open dialog 3 In the Files of Type drop down list select maplet 4 Navigate to the location of the maplet file and select the file 5 Click Open Maple Document To launch a Maplet appl
215. ies 320 7 Creating Mathematical Documents Drawing To draw with the pencil tool in the canvas 1 From the Drawing icons select the pencil icon 2 Click and drag your mouse in the canvas to draw lines Release the mouse to complete the drawing To adjust the color of drawing tools 1 From the Drawing icons select the Drawing Outline icon See Figure 7 16 2 Select one of the color swatches available or select the color wheel RGB ranges or eye dropper icon at the bottom of the dialog and customize the color to your preference 3 After selecting a new color draw on the canvas using the pencil icon and notice the new color Figure 7 16 Drawing Outline Color Icon In your document there are three plots two of which are 2 D plots that can be drawn on All of the information in the table you made in the previous section could be drawn onto the plot putting the information in a more concise layout 7 5 Canvas 321 Consider one of the plots from the table Click on the plot and notice that the Plot toolbar is open However the Drawing toolbar is also available Click on Drawing to see the toolbar Select the Text icon IT and click on the plot Enter the expression f x in one text area and its derivative in another as shown You can move the text areas around on the plot so that they indicate the correct lines For details on the rest of the drawing features refer to the DrawingTools help page
216. in loop The general for from loop has the following syntax for counter from initial by increment to final While conditional expression do statement sequence end do The general for in loop has the following syntax for variable in expression while conditional expression do statement sequence end do After testing the loop bound condition at the beginning of each iteration of the for loop Maple evaluates conditional_expression e If conditional_expression evaluates to false or FAIL Maple exits the loop e If conditional_expression evaluates to true Maple executes statement_sequence Infinite Loops You can construct a loop for which there is no exit condition for example a while loop in which the conditional_expression always evaluates to true This is called an infinite loop Maple indefinitely executes an infinite loop unless it executes a break quit or return statement or you interrupt the computation For more information refer to the break 2quit return and interrupt help pages Additional Information For more information on the for statement and looping refer to the do help page 380 9 Basic Programming 9 3 Iterative Commands Maple has commands that perform common selection and repetition operations These commands are more efficient than similar algorithms implemented using library commands Table 9 2 lists the iterative commands Table 9 2 Iterative Commands Command add Compute num
217. in the tutor 11 53 1 103 io io 10 145 445 5i 53 Change the matrix 7 Click the Close button to return the solution to your document L3 p Z 4 2 1 5 7 0 3 5 4 7 3 6 3 linear solve tutor For more information on linear systems and matrices see Linear Algebra page 159 3 Worksheet Mode The Worksheet mode of the Standard Worksheet interface is designed for e Interactive use through Maple commands which offers advanced functionality and cus tomized control not available using context menus or other syntax free methods e Programming using the powerful Maple language Using Worksheet mode you have access to all of the Maple features described in Chapter 1 and most of those described in Chapter 2 including e Math and Text modes e Palettes e Context menus e Assistants and tutors For information on these features see Chapter 1 Getting Started page 1 and Chapter 2 Document Mode page 63 Note Using a document block you can use all Document mode features in Worksheet mode For information on document blocks see Document Blocks page 51 Note This chapter and the following chapters except Chapter 7 were created using Worksheet mode 3 1 In This Chapter Suppressing Output 2 D and 1 D Math Input Input Separators Commands page 82 Thousands of routines for The Maple Library performing computations and other operations Top Level Commands Package Commands
218. including those described in Table 4 3 Table 4 3 Overview of Solution Methods for Important Equation Types Equation Type Solution Method LinearAlgebra LinearSolve command 112 4 Basic Computations Note Many solve operations are available in context menus and as task templates Tools Tasks Browse Most of this section focuses on other methods Solving Equations and Inequations Using Maple you can symbolically solve equations and inequations You can also solve equations numerically To solve an equation or set of equations using context menus 1 Right click for Macintosh Control click the equations 2 From the context menu select Solve or Solve Numerically See Figure 4 2 4 4 Solving Equations 113 Copy as MathML Numeric Formatting Explore Apply a Command pproximate I Combine b Complete Square b Cross Multiply Differentiate Evaluate at a Point Expand Integrate b Left hand Side Manipulate Equation Map Command Onto Move to Left Move to Right Negate Relation Right hand Side Sequence I Simplify b Solve p Isolate Expression For b Test Relation Numerically Solve Numerically Solve From point Dui Obtain Solutions For Integral Transforms b RE ii Solve explicit l Solve general solution Solve For Variable p Figure 4 2 Context Menu for an Equation In Worksheet mode Maple inserts a calling sequence that solves the equation followed by the solut
219. ine and New user resources including the Maple Tour from within Maple and the Maple Portal Examples Online help Maple web site resources 1 1 Introduction to Maple Working in Maple With Maple you can create powerful interactive documents The Maple environment lets you start solving problems right away by entering expressions in 2 D Math and solving these expressions using point and click interfaces You can combine text and math in the same line add tables to organize the content of your work or insert images sketch regions and spreadsheets You can visualize and animate problems in two and three dimensions format text for academic papers or books and insert hyperlinks to other Maple files web sites or email addresses You can embed and program graphical user interface components as well as devise custom solutions using the Maple programming language 1 1 Introduction to Maple 3 File Edit view Insert Format Table Drawno Plot Sprescsheet Tools Window Help DZ2aSS XWA oe TP BE es MI OHO amp QRS Plot Animation G C 2D Math v Times New Roman y 2 v B U eg ap er a aT x 7 x 1 F 1 7 ae dis 2 Inla x 8x 25x 200 cost 520 sin x 0 5 x 0 0 logig log a aae g ay ee apap 4 ee x 7 8 x 3 sin a cos a tan 2 fla Jie a Recall that tae Bay a b z x a x x lt 0 X XSA The graph of
220. ing text in the document 1 Highlight the text that you want to make a hyperlink 2 From the Format menu select Convert To and then Hyperlink 3 In the Hyperlink Properties dialog box the Link Text field is grayed out since the text region you highlighted is used as the link text This is demonstrated in Figure 7 18 The highlighted text region Diff is grayed out 4 Specify the hyperlink Type and Target as described in the appropriate following section To insert a text or image hyperlink into the document 1 From the Insert menu select Hyperlink 2 In the Hyperlink Properties dialog box enter the Link Text Optionally use an image as the link Select the Image check box and click Choose Image for the file In mw files the image appears as the link You can resize the image as necessary Click and drag from the corners of the image to resize 3 Specify the hyperlink Type and Target as described in the appropriate following section 7 6 Hyperlinks 325 Linking to a Web Page To link to a Web page 1 In the Type drop down list select URL 2 In the Target field enter the full URL for example http www maplesoft com 3 Click OK Linking to an Email Address To link to an email address 1 In the Type drop down list select Email 2 In the Target field enter the email address 3 Click OK Note For information about email hyperlinks in the Classic Worksheet interface see Worksheet Compatibility page
221. ing the ODE Analyzer Assistant 1 Click the Solve Symbolically button 2 In the Solve Symbolically window Figure 4 5 you can specify the method and relevant method specific options to use for solving the problem 3 To compute the solution click the Solve button 124 4 Basic Computations H Solve Symbolically Method Hutput Default timelimit s 60 Large Display l i ga g t ae cos t q os Use Classification Methods Integrate ato Explicit ato Ei Transforms laplace Truncated Series order Flot Options l Expansion point Show Maple commands soll dsolve diff diff iqit aa oes fad PE ee So E E Ss Dig 0 0 iqit plot 4 3 cos t 4 3 c Figure 4 5 ODE Analyzer Assistant Solve Symbolically Dialog When solving numerically or symbolically you can view a plot of the solution by clicking the Plot button e To plot the solution to a symbolic problem all conditions and parameters must be set e To customize the plot click the Plot Options button to open the Plot Options window To view the corresponding Maple commands as you solve the problem or plot the solution select the Show Maple commands check box You can control the return value of the ODE Analyzer using the On Quit Return drop down list You can select to return nothing the displayed plot the computed numeric pro cedure for numeric solutions the solution for symbolic solutions or the M
222. ing the eval command gt equation4 sin x cos x V solve equation4 R 4 10 gt eval equation4 x 4 10 f2 2 4 11 2 A 2 gt equations cos z gt fsolve equationS 2 498755763 4 12 gt eval equations z 4 12 8003983544 0 8003983540 4 13 For more information see Substituting a Value for a Subexpression page 359 Assigning the Value of a Solution to a Variable To assign the value of a solution to the corresponding variable as an expression use the assign command For example consider the numeric solution in 4 9 z 98 98037599 found using the starting value z 100 120 4 Basic Computations gt assign 4 9 gt Z 98 98037599 Creating a Function from a Solution The assign command assigns a value as an expression to a name It does not define a function To convert a solution to a function use the unapply command Consider one of the solutions for q to the equation g rst a 5 gt solutions gt solvel eg 1 5 4 F 1 l JV14 4rs 20 1 14 1 4rs 20r r r solutions gt pa gt f unapply solutions 1 r s 1 47 s 20 7 P Here solutions 1 selects the first element of the list of solutions For more information on selecting elements see Accessing Elements page 338 You can evaluate this function at symbolic or numeric values 4 4 Solving Equations 121 gt f x y 4
223. ints of horizontal range i v2 x coordinates and y coordinates Go to a Bookmark You can automatically move the cursor to the location of the bookmark in the active docu ment 1 From the Edit menu select Go To Bookmark The Go To Bookmark dialog opens with the current bookmarks listed 2 Select the bookmark parameters and click OK The cursor moves to the bookmark at the beginning of the Parameters section For more information refer to the gt bookmarks help page 7 7 Embedded Components You can embed simple graphical interface components such as a button in your document These components can then be associated with actions that are to be executed For example the value of a slider component can be assigned to a document variable or a text field can be used to input an equation Adding Graphical Interface Components The graphical interface components can be inserted by using the Components palette Figure 7 21 or by cutting copying and pasting existing components to another area of the document Although copied components have most of the same characteristics they are distinct By default palettes are displayed when you launch Maple If palettes are not visible use the following procedure 330 e e 7 Creating Mathematical Documents 1 From the View menu select Palettes 2 Select Expand Docks 3 If the Components palette is not displayed right click Control click for Macintosh the palette dock Fr
224. ion page 366 or refer to the uneval help page See also Unassigning a Name Using Unevaluation Quotes page 368 Unassigning all names The restart command clears Maple s internal memory The effects include unassigning all names For more information refer to the restart help page Note To execute the examples in this manual you may be required to use the unassign or restart command between examples Valid Names A Maple name must be one of the following e A sequence of alphanumeric and underscore _ characters that begins with an alphabet ical character Note To enter an underscore character in 2 D Math enter a backslash character followed by an underscore character that is _ e A sequence of characters enclosed in left single quotes C Important Do not begin a name with an underscore character Maple reserves names that begin with an underscore for use by the Maple library Examples of valid names ca e al e polynomial e polynomiall divided_by_polynomial2 e 2a e xy 3 10 Equation Labels 97 3 10 Equation Labels Maple marks the output of each execution group with a unique equation label Note The equation label is displayed to the right of the output gt sin x dx cos x 3 4 Using equation labels you can refer to the result in other computations gt 3 4 dx sin x 3 5 Displaying Equation Labels Important By default equation labels are displayed
225. ions If you select Solve Maple computes exact solutions 114 4 Basic Computations gt i y L 2 4 3 gt solve 4 3 ro a VIB fx VT 4 4 If you select Solve Numerically Maple computes floating point solutions y gt x 12 a E ae x 12 4 5 gt fsolve 4 5 x 2 063602674 x 2 492174103 4 6 For information on solving equations and inequations symbolically using the solve command see the following section For information on solving equations numerically using the fsolve command see Numerically Solving Equations page 117 Symbolically Solving Equations and Inequations The solve command is a general solver that determines exact symbolic solutions to equations or inequations The solutions to a single equation or inequation are returned as an expression sequence For details see Creating and Using Data Structures page 337 If Maple does not find any solutions the solve command returns the empty expression sequence gt solve x 3x 14 0 In general solve computes solutions in the field of complex numbers To restrict the problem to only real solutions see Restricting the Domain page 143 It is recommended that you verify the solutions returned by the solve command For details see Working with Solutions page 119 4 4 Solving Equations 115 To return the solutions as a list enclose the calling sequence in brackets gt solve x7 T
226. ip account gives you access to thousands of free Maple resources and MaplePrimes which is an active web community for sharing techniques and experiences with Maple and related products To sign up for a free Maplesoft com membership account visit http www maplesoft com members sign up form aspx The MapleCloud is integrated with several of these online features so it is strongly recommended that you use a Maplesoft com membership account 424 11 Input Output and Interacting with Other Products Index Symbols toolbar icon 69 toolbar icon 69 347 178 H 171 T 171 amp x 171 95 366 384 gt 95 169 1 D Math 81 switching to 2 D 81 2 D Math 80 converting to 1 D 81 entering 5 shortcuts 7 switching to 1 D 81 81 82 1 144 94 Sl 82 lt gt 159 161 gt 80 is help topic 55 167 338 339 6 110 entering 110 _ 96 entering 96 _ZN 116 96 338 162 116 144 A about command 144 abs command 107 absolute value 107 add word to your dictionary 334 add command 381 additionally command 145 algebra 157 linear 173 polynomial 150 algsubs command 360 alignment format 286 American spelling spellcheck 332 and operator 370 angle brackets 159 161 206 angles 356 animations creating 273 customizing 277 Application Center 60 applications sample documents 58 apply character styles 289 paragraph
227. isted NO Click Browse and navigate to the directory in which Maple 1s installed In the Excel directory select the WMIMPLEX xla file Click OK 3 Select the Maple Excel Add in check box 4 Click OK More information is available in the Using Maple in Excel help file within Excel To view this help file 1 Enable the add in 2 From the View menu select Toolbars and then Maple 3 On the Maple toolbar click the Maple help icon 4 OpenMaple OpenMaple is a suite of functions that allows you to access Maple algorithms and data structures in your compiled C Java or Visual Basic programs This is the reverse of ex ternal calling which allows access to compiled C Fortran 77 and Java code from Maple To run your application Maple must be installed You can distribute your application to any licensed Maple user For additional terms and conditions on the use of OpenMaple refer to the extern OpenMapleLicensing txt file in your Maple installation For more details on using OpenMaple functions refer to the 0penMaple help page MapleSim MapleSim is a complete environment for modeling and simulating multi domain engin eering systems During a simulation MapleSim uses the symbolic Maple computation engine to generate the mathematical models that represent the system behavior Because both products are tightly integrated you can use Maple commands and technical document features to edit manipulate and analyze a
228. its value For example gt cos a For information on Maple evaluation rules see Evaluating Expressions page 359 Mathematical Functions To define a function assign it to a name For example define a function that computes the cube of its argument 3 9 Names 95 3 gt cube x x For information on creating functions see Define a Mathematical Function page 66 gt cube 3 cube 1 666 27 4 624076296 Note To insert the right arrow enter the characters gt In 2 D Math Maple replaces gt with the right arrow symbol In 1 D Math the characters are not replaced For example define a function that squares its argument gt square x gt x 2 gt square 32 1024 For more information on functions see Functional Operators page 343 Protected Names Protected names are valid names that are predefined or reserved If you attempt to assign to a protected name Maple returns an error gt sin 2 Error attempting to assign to sin which is protected For more information refer to the type protected and protect help pages Unassigning Names The unassign command resets the value of a name to itself Note You must enclose the name in right single quotes gt unassign a 96 3 Worksheet Mode cf Right single quotes unevaluation quotes prevent Maple from evaluating the name For more information on unevaluation quotes see Delaying Evaluat
229. its package 128 environments 132 extensibility 134 UseSystem command 134 UsingSystem command 133 Units palettes 75 131 universal gravitational constant 135 UNIX command complete 7 context menus 39 unwith command 85 URL adding hyperlink to 325 V variables 65 variance 141 VariationalCalculus package 187 Vector constructor vectorfield attribute 186 data structure 159 vector fields 186 vector spaces basis 173 VectorCalculus package description 87 VectorCalculus package 186 Student version 187 vectors 343 arithmetic 169 column 162 context menus 171 cross product 171 data type 165 defining 161 efficiency 164 filling 165 large 163 multiplication 169 row 162 165 scalar multiplication 170 selecting entries 166 shape 165 transpose 171 View menu in help system 57 markers 52 Volume Gauge component 391 W Web page adding hyperlink to 325 Web site access to Maple help pages 61 Application Center 60 198 MaplePrimes 61 Student Center 199 Student Help Center 60 Teacher Resource Center 60 Technical Support 61 Training 60 Welcome Center 60 Welcome Center 60 while loops 376 Windows command complete 7 context menus 39 with command 84 worksheet adding hyperlink to 325 Worksheet Environment 3 Worksheet Migration Assistant 37 Worksheet mode 63 79 worksheets running 9 write to files 413 X xor operator 370 Z zero recognition
230. ksheet mode supports the features available in Document mode described in this chapter For information on using Worksheet mode see Chapter 3 Worksheet Mode page 79 Note To enter a Maple input prompt while in Document mode click in the Maple toolbar Important In any Maple document you can use Document mode and Worksheet mode Interactive document features include e Embedded graphical interface components like buttons sliders and check boxes e Automatic execution of marked regions when a file is opened e Tables e Character and paragraph formatting styles e Hyperlinks These features are described in Chapter 7 Creating Mathematical Documents page 281 Note This chapter and Chapter were created using Document mode All of the other chapters were created using Worksheet mode 2 3 Entering Expressions Chapter 1 provided an introduction to entering simple expressions in 2 D Math see Entering Expressions page 19 It is also easy to enter mathematical expressions such as 2 3 Entering Expressions 65 xy lt Q Piecewise continuous functions x 0 x 0 x 8 ie alee Limits x lim qx E gt l Continued fractions y 2 1 24 2 oe Sree and more complex expressions Mathematical expressions can contain the following objects Numbers integers rational numbers complex numbers floating point values finite field elements 1 0 Operators L4 lim e
231. l Problem Solving 6 Click Return Steps to close the dialog and re x 7 mi j2 i 4 1 x turn all of the steps to the Maple document manipulate equation w tiir 4 17 46 4 3 x 1 4 x 4 0 x 7 6x 24x 18 0 pl I x 3 T Ctrl drag the factored form of the original _ u k 30 solutions for x 13 equation to a new document block region 8 Right click and select Solve Obtain Solu tions for x Instant Solution To apply an instant solution to this problem use context menus 1 Ctrl drag the equation x 7 ie 17 EA 1 4a new document block region 2 Right click the expression and select Solve gt x 7 x 1 4 x 1 x Obtain Solutions for x solutions for x n G l Step by step Interactive Solution This equation can also be solved interactively in the document by applying context menu operations or commands one step at a time 1 Ctrl drag the equation x 7 x 1 7 4 x 1 x 4 7 toa blank document block region Group all terms on the right 2 Right click this equation and from the context menu select Move to Right Expand the expression on the right hand side 3 Right click on the result and from the context menu select Expand Use Maple s factor command on the resulting right hand side 4 Right click on the result and select Right hand Side 5 Right click on the
232. le gt Mean Z Ny W Statistical Computations In addition to basic functions like mean median standard deviation and percentile the Statistics package contains commands that compute for example the interquartile range and hazard rate Example 1 Interquartile Range Compute the average absolute range from the interquartile of the Rayleigh distribution with scale parameter 3 gt InterquartileRange Rayleigh 3 3 36 J In 2 ore To compute the result numerically e Specify the numeric option 5 6 Statistics 195 gt InterquartileRange Rayleigh 3 numeric 2 719744818 Example 2 Hazard Rate Compute the hazard rate of the Cauchy distribution with location and scale parameters a and b at an arbitrary point t gt HazardRate Cauchy a b t l i 7 1 arctan Fi wo i ia eo i b IU You can specify a value for the point t gt HazardRate Cauchy a b gt i Ey a gt r a m 7 IU gt t You can also specify that Maple compute the result numerically gt HazardRate Cauchy 10 1 imumeric 0 003608801460 For more information refer to the Statistics DescriptiveStatistics help page Plotting You can generate statistical plots using the visualization commands in the Statistics package Available plots include e Bar chart e Frequency plot 196 e 5 Mathematical Problem Solving e His
233. le borders on mouse pointer roll over and annotations The dialog is accessed from the View Show Hide Contents menu Using the Show Contents Dialog A check mark beside the item indicates that all document elements of that type are displayed for the current document See Figure 7 9 Show Contents Components Spreadsheets Input Output Graphics Markers Section Boundaries Execution Group Boundaries Hidden Table Borders Annotation Markers Figure 7 9 Show Contents Dialog 7 2 Document Formatting 299 1 From the View menu select Show Hide Contents The Show Contents dialog opens with all items selected for display 2 Clear the check box associated with the document components or markers to hide them Note By clearing the Input check box only Maple Input and 2 D Math input that 1s 2 D Math content that has been evaluated are hidden Clearing the Graphics check box ensures that a plot an image or the Canvas inserted in the document by using the Insert menu option is also hidden Command Output Versus Inserted Content Output is considered an element that results from executing a command Inserted components are not considered output Consider the following examples The plot resulting from executing the plot sin command is considered output e To show a plot from the plot sin command select both the Output and Graphics check boxes in the Show Contents dialog If you insert a plot by using the
234. le illustrates the vertical alignment options The baseline option is useful for aligning equations across multiple cells within a row of a table 7 4 Tables 313 For example set the Row alignment to Baseline for all rows and set the Column alignment to Center for all columns de ahaa i fd al Plot off x and af x I Tre 2 l R T f 3 x 7 57 f 1X sin cx e cos eox wett Ssin ox e i Te amn ix 8 s x cos x dx Cell Color You can set the background color of any cell or collection of cells to be any color This coloring is independent of any highlighting or text color that may also be applied To change the color of a cell place the cursor in the cell then from the Table menu select Cell Color In the Select A Color dialog choose a color from the swatches the color wheel or RGB See the 7DrawingTools help page for details on color selection 314 7 Creating Mathematical Documents For example select the first row of the table and apply a light blue color This sets the header off from the content below sinf x e1722 cos wx we ts sinf w x e72 i 3 amn Xx sin x cos x da Controlling the Visibility of Cell Content The Table Properties dialog includes two options to control the visibility of cell content These options allow control over the visibility of Maple input and execution group bound aries Thus these elements can be hid
235. le objects To sort a polynomial 1 Right click Control click for Macintosh the polynomial 2 The context menu displays From the Sorts menu select e Single variable and then the unknown e Two variable or Three variable Pure Lexical or Total Degree and then the sort priority of the unknowns See Figure 5 1 Copy as MathML Paste Explore Apply a Command Assign to a Mame Coefficients Collect Combine Complete Square Differentiate Differentiate Implicitly Evaluate at a Point Factor Integrate Limit Sequence Series Simplify Solve Complex Maps Constructions Conversions Integer Functions Integral Transforms Language Conversions Optimization Plats Sorts Units 2 D Math T F F F F Single variable Two variable P Figure 5 1 Sorting a Polynomial Using a Context Menu Maple sorts the polynomial 5 2 Algebra Pure Lexical P Total Degree Ka Y YX 155 In Worksheet mode Maple inserts the calling sequence that performs the sort followed by the sorted polynomial a 4 7 9 gt x ty xy 156 5 Mathematical Problem Solving gt sort x 3 y 3 x42 y 2 ly x plex ytyxrtx You can use context menus to perform operations on 2 D Math content including output For more information see Context Menus page 70 for Document mode or Context Menus page 89 for Worksheet mode Collecting Terms To collect the terms of polynomial use the collect command 3 y
236. lickButtant command AE J x Palette Pane Command Pane Properties Pane Figure 10 7 Maplet Builder Interface The Maplet Builder is divided into four different panes e The Palette pane displays palettes which contain Maplet elements organized by category For a description of the elements see the MapletBuilder Palette help page The Body palette contains the most popular elements e The Layout pane displays the visual elements of the Maplet e The Command pane displays the commands and corresponding actions defined in the Maplet e The Properties pane displays the properties of an instance of a defined element in the Maplet 404 10 Embedded Components and Maplets Example 3 Design a Maplet Using the Maplet Builder In this example shown in Figure 10 8 the Maplet user enters a function and plots the result 3 Maplet Figure 10 8 Image of the Maplet Y Body lr Ekee GEk Define the number of rows in the Maplet 1 In the Properties pane a In the drop down list select Box Column1 b Change the numrows field to 2 Ele bl Button element Label element Plot element TextField element 10 5 Authoring Maplets 405 Add a plot region to row 1 2 From the Body palette drag the Plotter element to the first row in the Layout pane Y Dialog z iS W hieny EBEE Add columns to row 2 3 In the Properties p
237. lt option values A expression with a singularity is one such example 6 3 Customizing Plots 265 l gt plot x 5 5 oy 4 x 10 3 x 10 1 x 10 In the previous plot all interesting details of the plot are lost because there is a singularity at x 1 The solution is to view a narrower range for example from y 0 to 7 Alter the y axis range 1 Right click the plot region Select Axes and then Properties 2 In the Axes Properties dialog click the Vertical tab 3 De select the Use data extents check box and enter 0 and 7 in the Range min and Range max text regions respectively 4 Click Apply to view the changes or OK to return to the document Change the color 5 Place the mouse pointer on the curve and right click Control click Macintosh Note The curve is selected when it becomes highlighted Select Color and then Green 266 6 Plots and Animations Change the line style 6 Select Style and then Point 3 D Plot Options By default Maple displays the graph as a shaded surface with a wireframe and scales the plot to fit the window To change these options use the context menu gt plotsdl 5 x 10 10 y 5 5 x y Maple has many preselected light source configurations Change the style 1 Right click the plot region Select Style Surface Apply a light scheme 2 Select Lighting Light 1 Change the color 3 Select Color Z Gra
238. lutions 143 numerically 117 symbolically 114 transcendental 116 errors quantities with 139 Euclidean algorithm 158 Index 429 eval command 359 385 evalb command 362 evalc command 362 evalf command 104 116 137 140 361 with Int command 185 with Limit command 176 evaln command 366 evaluation boolean expressions 362 complex expressions 362 delaying 366 levels of 365 Maple expressions 359 of expression at a point 359 output below 68 71 output inline 67 70 updated computations 68 exact computation 103 numbers 102 quantities converting to floating point 104 example worksheets copy 57 execution group 81 execution groups 19 expand command 355 document block 302 execution group 303 series 182 Exploration Assistant 43 exponents entering 6 export 386 to HTML 416 to LaTeX 416 to Maple input 417 to Maple T A 420 to Maple text 417 to Maplet application 417 to other formats 419 to PDF 417 430 Index to plain text 417 to Rich Text Format 417 worksheets 416 exporting embedded components 392 expression sequences 114 338 creating 380 expressions 65 337 adding 381 evaluating 359 manipulating 353 multiplying 381 right click 40 versus functional operators 344 factor integers 106 polynomials 157 QR factorization 173 factor command 157 354 factored normal form 357 factorial command 108 FAIL 370 377 false 370 377 Faraday constant
239. lying Expressions Selecting Expression Operands Mapping a Command over a Set or List Mapping a Binary Command over Two Lists or Vectors Procedures page 383 Maple programs Defining and Running Simple Procedures Procedures with Inputs Procedure Return Values Displaying Procedure Definitions Displaying Maple Library Procedure Definitions Modules Programming in Documents page 387 Dis Code Edit Region play methods for Maple code Startup Code Document Blocks 369 370 9 Basic Programming 9 2 Flow Control Two basic programming constructs in Maple are the if statement which controls the condi tional execution of statement sequences and the for statement which controls the repeated execution of a statement sequence Conditional Execution if Statement You can specify that Maple perform an action only if a condition holds You can also perform an action from a set of many depending on which conditions hold Using the if statement you can execute one statement from a series of statements based on a boolean true false or FAIL condition Maple tests each condition in order When a condition is satisfied Maple executes the corresponding statement and then exits the if statement Syntax The if statement has the following syntax gt if conditional expression then statement sequencel elif conditional expressions then statement sequences elif conditional expressions then s
240. manipulate matrices and vectors compute standard operations perform queries and solve linear algebra problems Table 5 7 lists some LinearAlgebra package commands For a complete list refer to the LinearAlgebra Details help page Table 5 7 Select LinearAlgebra Package Commands Basis Return a basis ora vector space SOS S S S S Dimension Determine the dimension of a matrix ora vector MatrixInverse Compute the inverse of a square matrix or pseudo inverse of a non square matrix QRDecomposition Compute the QR factorization of a matrix Random Matrix Construct a random matrix Sylvester Matrix Construct the Sylvester matrix of two polynomials For information on arithmetic operations see Matrix Arithmetic page 169 For information on selecting entries subvectors and submatrices see Accessing Entries in Matrices and Vectors page 166 Example Determine a basis for the space spanned by the set of vectors 2 13 15 7 2 13 5 4 9 Express the vector 25 4 9 with respect to this basis gt with LinearAlgebra 174 5 Mathematical Problem Solving gt vi lt 2 13 15 gt v2 lt 7 2 13 gt v3 lt 5 4 9 gt Find a basis for the vector space spanned by these vectors and then construct a matrix from the basis vectors gt basis Matrix Basis vl v2 v3 2 y 3 basis 13 2 4 15 3 9 To express 25 4 9 in this basis use the LinearSolve command
241. mation about names with assumptions see Assumptions on Variables page 144 RootOf Structure The solve command may return solutions for example to higher order polynomial equations in an implicit form using RootOf structures gt solve x 2x 3x 2 1 Root0f Z 24 2 242 74 2 index 1 RootOf Z 2 2 242 74 2 index 2 RootOf A 2 2 2 2 Z 2 index 3 Rootof Z Z 2 7 4 7 2 Z 2 index 4 These RootOf structures are placeholders for the roots of the equation 2 7 27 2z 2 The index parameter numbers and orders the four solutions Like any symbolic expression you can convert RootOf structures to a floating point value using the evalf command 4 4 Solving Equations 117 gt evalf 4 7 1 0 984001051867989 1 52659083388421 I 0 484001051867989 0 609947 140486231 I 0 484001051867989 0 609947140486231 I 0 984001051867989 1 52659083388421 I Some equations are difficult to solve symbolically For example polynomial equations of order five and greater do not in general have a solution in terms of radicals If the solve command does not find any solutions it is recommended that you use the Maple numerical solver fsolve For information see the following section Numerically Solving Equations For more information on the solve command including how to solve equations defined as procedures and how to find parametric solutions refer to the solve details help page For info
242. matting Explore Apply a Command Assign to a Mame Integer Factors Next Prime be Test Primality Integer Functions b Units a Number Theory Functions b Figure 4 1 Context Menu for an Integer The result of applying Integer Factors is shown gt 9469629 9469629 4 1 gt ifactor 4 1 3 13 17 23 4 2 Maple inserts the command ifactor using an equation label reference to the integer 946929 For more information on equation labels see Equation Labels page 97 For more information on using context menus in Worksheet mode see Context Menus page 89 For information on using context menus in Document mode see Context Menus page 70 Maple has many other integer commands including those listed in Table 4 1 Table 4 1 Select Integer Commands abs absolute value displays in 2 D math as _ a 108 4 Basic Computations T l iuo ouotient of integer division SSOS S moa modular arithmetic See Finite Rings and Fields page 1091 gt iquo 209 17 12 gt irem 209 17 5 gt igcd 2024 4862 22 gt iroot 982523 4 3 For information on finding integer solutions to equations see Integer Equations page 126 Non Base 10 Numbers and Other Number Systems Maple supports e Non base 10 numbers e Finite ring and field arithmetic e Gaussian integers Non Base 10 Numbers To represent an expression in another base use the convert command 4 3 Integ
243. mmand completion and over 20 palettes using Maple has never been so easy This chapter is designed to show several ways to solve the same problem in Maple Throughout these examples you will need to insert new document block regions This is done through the Format menu by se lecting Create Document Block Also these examples only use the keyboard keys needed for a Windows operating system Refer to Shortcut Keys by Platform page xiv for the keys needed for your operating system Example 1 Graph a Function and its Derivatives On the interval 1 2 graph f f and f for f x xcos x We solve this problem using the following methods e Solution by Context Menus page 213 Solution by Tutor page 215 e Access the Tutor from a Task Template page 217 Solution by Context Menus 1 Enter the expression x cos x x cos x Make a copy of the expression and calculate the differentiate w r t x a x cos x derivative cos x xsin x 2 Insert anew document block region by select ing from the Format menu Create Document Block 3 Highlight the original expression Ctrl drag the expression to the new document block 4 Right click the expression and select Differ entiate x Make a copy of the derivative and calculate differentiate w r t x the second derivative cos x xsin x 2 sin x xcos x 5 Insert a new document block and Ctrl drag the derivative to the document block 6 Right
244. move the camera close to or away from the surface Refer to the viewpoint help page for information on the available options 274 6 Plots and Animations To animate the following examples click the plot object and then click the play button in the Animation context bar Example 1 Moving the Camera Around a 3 D Plot In the following example a pre defined path circleleft moves the camera in a counter clockwise circle around the plot surface gt plot3a 1 3 sin y x 1 2 n y 0 1 coords spherical style patch viewpoint circleleft Example 2 Specifying a Path to Move the Camera Towards and Around a 3 D Plot In the following example a camera path is specified to zoom into and view different sides of the plot surface 6 6 Playing Animations 275 gt plot3d sin x y x 1 1 y 1 1 shading xyz viewpoint path 50 x 90 cos x 100 sin x x 2 2 2 SN lt SS 6 6 Playing Animations Animation Context Bar To run the animation click the plot to display the Animate context bar Table 5 6 Animation Options Frame ation SI a bi View the next frame in the animation Current Current Frame 20 Slider control for viewing individual Frame frames of an animated plot 276 6 Plots and Animations Forward g Forward Play the animation forward Oscillate Tei T Oscillate Play the animation forward and backward Backward Backward Pla
245. mple Example 6 Enter Text and 2 D Math in the Same Line Using Toolbar Icons Enter the following sentence Evaluate 5 3 Be eae 31x dy and write in simplest terms l To enter this sentence 1 Select the Text icon and enter Evaluate 2 Select the Math icon 3 From the Expression palette select the a C 20 Math 7 dx Brat 7 dx definite integration template 2 The expression is displayed with the first place holder highlighted 4 With the first placeholder highlighted enter 1 then press Tab C 2D Math 5 Enter 5 and press Tab to highlight the in s s tegrand region mahae F ax l 6 Enter 3x 2 and press the right arrow to leave the superscript position Drawing C 2D Math Y Times Me 1 Enter 2 Ey 5 Evaluate Jx J dx FaN l l 32 e 1 Getting Started 8 Press the Space bar for implicit multiplica tion Enter sqrt and press Esc to show the command completion options Maple displays a pop up list of exact matches Select the al 5 Evaluate 3a 42 sort dx 2 l sgrt me symbol with the x placeholder selected Al ternatively select the square root symbol from the Expression palette Drawing Plot Anim K C 2D Math w j i Times New Roman square root symbol J x Maple inserts the Anima 9 Enter x then press the right arrow to leave yo44 GED prawing Pio the square root region e k C 2D Math Times New Roman
246. must apply a character style or character formatting e A paragraph style controls all aspects of a paragraph s appearance such as text alignment line spacing and indentation In Maple each paragraph style includes a character style 7 2 Document Formatting 289 Style Management P Annotation Title Create Character Style P Author P Bullet Item Create Paragraph Style P Dash Item p Modify P Diagnostic Delete P Error P Heading 1 P Heading 2 P Heading 3 P Heading 4 Figure 7 4 Style Management Dialog Applying Character Styles By using the drop down list in the document context bar you can apply e Existing Maple character styles e New styles that you have created through the Style Management Figure 7 4 and Character Style Figure 7 5 dialogs To apply a character style to text in your document 1 Select the text to modify 2 In the styles drop down list in the context bar of your document select an appropriate character style All character styles are preceded by the letter C The selected text now reflects the attributes of the character style you have chosen CC tet C Code al C Dictionary Hyperlinl C Equation Label L Hyperlink C Maple Input C Maple Input Placehe C Page Number C Text 3 Optional If necessary you can remove this style From the Edit menu select Undo 290 e 7 Creating Mathematical Documents Creating and Modifying Character Styles You can
247. n T to The answer to enter plain text Enter The answer to Note these instructions are for Worksheet mode 2 Click the input prompt icon to enter Maple The answer to gt sin x dx gt 3 Again click the text icon to insert the rest of the The answer to text is and then enter another input prompt icon gt sinf x dx Make sure to put spaces around all of the text so the l sentence displays properly is gt commands Enter sin x dx and then press Enter to execute the command 54 1 Getting Started 4 To display the same output again use the value The answer to command and an equation label This allows you to gt sin x dx insert text between the input and output of a single cos x command there are really two commands Lis gt value 1 3 gt 5 To finish the sentence click the text icon in the The answer to last execution group and enter a period gt sin x dx cos x Enter and execute the command as shown cos x cos x is lg vaiwe 1 3 L 6 Select the entire sentence then from the Format gt menu select Create Document Block By default only the text and output remains visible and output is is centered on a new line The answer to 7 To display the text and output on one line place 54 The answer to cog x is cos x the cursor in the document block From the View menu select Inline Document
248. n on indexing methods refer to the 9rtable_ indexing help page Large Arrays Only one and two dimensional Arrays with at most 10 indices in each dimension display in the document Larger Arrays display as a placeholder 342 8 Maple Expressions gt Array 0 100 0 100 Array Data Type anything Storage rectangular Order Fortran_order To view large Arrays e Double click the placeholder The Matrix Browser displays the Array For more information see Viewing Large Matrices and Vectors page 162 Tables Tables are conceptually an extension of the Array data structure but the table data structure is implemented using hash tables Tables can be indexed by any values not only integers Defining Tables and Accessing Entries gt Greek table a a b B c y gt Greek b p You can also assign anything for example a list to each element gt Translation table one un uno two deux dos three trois tres gt Translation two deux dos For more information on tables refer to the table help page 8 2 Creating and Using Data Structures 343 Matrices and Vectors Matrices and Vectors are specialized data structures used in linear algebra and vector calculus computations 12 33 83 12 yi lt 2 14 gt For information on defining Matrices and Vectors see Creating Matrices and Vec tors page 159 gt My 486 334 gt v M 1
249. n this example we will enter gt 7 p i and evaluate the expression i 0 1 Place the cursor in a new document block In the Ex S e pression palette click the summation template Maple inserts the summation symbol with the range variable placeholder highlighted 28 e 1 Getting Started 2 Enter i and then press Tab The left endpoint placehold er is selected Notice that the color of the range placehold er has changed to black Each placeholder must have an assigned value before you execute the expression The i Tab key advances you through the placeholders of an inserted palette item 3 Enter 1 and then press Tab The right endpoint place holder is selected 4 Enter 10 and then press Tab The expression placehold er is selected 5 Enter 7 p Si For instructions on entering this type of expression see Example 1 Enter and Evaluate an Expression page 8 6 Press Ctrl Command for Macintosh to 0 evaluate the summation a i 2420 i Handwriting Palette The Handwriting palette provides another way to find and insert desired symbols easily 1 Draw the symbol with your mouse in the space provided m 2 Click the recognize button gt T available in the system See Figure 1 3 Maple matches your input against symbols 3 To view more symbols where indicated with a box around the result click the displayed symbol and choose one of the selections from the dro
250. nction is rotated about the horizontal or vertical axis Enter the function as an expression and specify the range gt sin x cos x 1 0 s sin x cos x 1 0 Fi 1 Calculate the volume of revolution pi pi 2 gt Shident Calculusi PolumeOfRevolution 1 16 Display the floating point value using the evalf command gt evalf 2 8 695 245151 3 Figure 5 16 Inserted Task Template 6 When a Task Template is inserted parameters are marked as placeholders denoted by purple font To navigate between placeholders press the Tab key After updating any parameters execute the command by pressing Enter Check for Instructions Help Page and Example Worksheet The help system provides command syntax information To access a help page 1 From the Help menu select Maple Help 2 In the search field enter volume of revolution and click Search The search results include the command help page the dictionary definition and the associated tutor help page 3 Review the calling sequence parameters and description in the Student Calcu lus1 VolumeOfRevolution help page 4 Copy the examples into your worksheet from the help system Edit menu select Copy Examples 5 7 Teaching and Learning with Maple 211 5 Close the Help Navigator 6 In your document from the Edit menu select Paste The examples are pasted into your document 7 Execute the examples and examine the results T
251. ndent Encloses the selection in a subsection For details refer to Sections page 295 Executes all commands in the worksheet or document Removes any section enclosing the selec a From the Format menu select Outdent From the Edit menu select Execute and then Worksheet 10 1 Getting Started Basic Usage Equivalent Menu Option or Command Executes a selected area From the Edit menu select Execute and then Selection Clears Maple s internal memory For de Enter restart tails refer to the restart help page Add and edit Maple code that is executed From the Edit menu select Startup Code each time the worksheet is opened For details refer to the startupcode help page Adjusts the display size of document Te de From the View menu select Zoom Factor content Note plots spreadsheets im and then a zoom size ages and sketches remain unchanged Opens the Maple help system For details From the Help menu select Maple Help refer to The Maple Help Sys tem page 54 For 1 D Math and text regions the Tab icon in the toolbar allows you to set the Tab key to move between placeholders or cells in a table or to indent text Table 1 3 Tab Icon Description Tab icon off Allows you to move between placeholders using the Tab key H Tab icon on Allows you to indent in the worksheet using the Tab key H ant The Tab icon is disabled when using 2 D Math Math mode and as such the Tab key allow
252. nents as follows Dial0 no changes Diall change the Value at Highest Position to 10 the Spacing of Major Tick Marks to 1 and the Spacing of Minor Tick Marks to 1 RotaryGauge0 change the Value at Highest Position to 40 the Spacing of Major Tick Marks to 5 and the Spacing of Minor Tick Marks to 1 Plot0 no changes 398 10 Embedded Components and Maplets e MathContainer0 change the Width in Pixels to 200 and the Height in Pixels to 45 Note the names of all of the components and close each dialog before moving on 3 Create actions for the components Components can perform actions when their values are changed so the code to execute needs to be in the dials That way whenever one of them 1s changed the other components are updated to reflect that change The following Maple commands retrieve the values of the parameters and display them in the other three components gt parameterl1 Do Dial0 gt parameter2 Do Diall gt Do SRotaryGauge0 parameter1 parameterz2 gt Do SPlot0 plot parameter2 x parameterl x 50 50 y 50 50 gt Do SMathContainer0 y parameter2 x parameterl1 4 Test the actions To test these commands first load the DocumentTools package with the following command gt with DocumentTools Execute the commands in the document and verify that the components you inserted are updated the gauge should change to the computed value a plot should appear in t
253. ng 140 element properties 141 rounding the error 140 scientific constants 141 with units 141 quick character formatting 284 paragraph formatting 286 quit statement 379 quo command 150 quotes double 347 left single 96 right single 95 366 unevaluation 366 quotient integer 108 R Radio Button component 390 random matrices 163 variables 193 randpoly command 158 range in plots 265 operator 168 rank 171 rational expressions entering 6 read from files 415 RealDomain package description 86 recurrence relation solving 127 reference equation labels 99 names 95 relational operators 370 rem command 150 remainder integer 108 remove command 381 repetition statements 373 reserved names 95 resources in help system 56 restart command 85 96 resultant command 158 return statement 379 values 385 rhs command 350 right single quotes 95 366 right click expressions 40 right hand side 350 RootOf structure 116 Index 437 roots command 158 of equations 116 Rotary Gauge component 391 row vector creating 165 rsolve command 127 running documents 9 worksheets 9 S saving a Maple Document 19 scatter plot 196 scientific constants 134 list 135 name 135 symbol 135 uncertainty 138 units 138 using 135 value 137 value and units 138 Scientific Constants Assistant 37 ScientificConstants package description 86 ScientificConstants package
254. nh x dx In 2 In e e In 1 D Math clicking the definite integration item inserts the corresponding command calling sequence 3 5 Context Menus 89 gt int x a b Specify the problem values using the Tab to move to the next placeholder and then press Enter gt int tanh x x 0 1 In 2 In e e Note Some palette items cannot be inserted into 1 D Math because they are not defined in the Maple language When the cursor is in 1 D Math input unavailable palette items are dimmed For more information on viewing and using palettes see Palettes page 22 in Chapter 1 3 5 Context Menus A context menu 1s a pop up menu that lists the operations and actions you can perform on a particular expression See Figure 3 2 gt 946929 946929 Copy as MathML Numeric Formatting Explore pply a Command 455ign to a Mame Integer Factors Next Prime k Test Primality Integer Functions b Units Number Theory Functions Figure 3 2 Integer Context Menu In Worksheet mode you can use context menus to perform operations on 2 D Math and output To use a context menu 90 3 Worksheet Mode 1 Right click Control click for Macintosh the expression The context menu is displayed 2 From the context menu select an operation Maple inserts a new execution group containing e The calling sequence that performs the operation e The result of the operation Ex
255. ns which are defined along the real line by probability density functions Maple supports many continuous distributions including the normal Student t Laplace and logistic distributions e Discrete distributions which have nonzero probability only at discrete points A discrete distribution is defined by a probability function Maple supports many discrete distribu tions including the Bernoulli geometric and Poisson distributions For a complete list of distributions refer to the Statistics Distributions help page You can define random variables by specifying a distribution in a call to the RandomVari able command gt with Statistics gt X RandomVariable Poisson A Find the probability distribution function for X For information on statistics computations see Statistical Computations page 194 gt PDF X t 5 ire Dirac t k k aa Adding Custom Distributions To add a new distribution specify a probability distribution in a call to the Distribution command 194 e 5 Mathematical Problem Solving 0 t lt 0 gt U Distribution PDF t gt 3 lt 3 QO otherwise To construct a piecewise continuous function in 1 D Math use the piecewise command for example t gt piecewise t lt 0 0 t lt 3 1 3 0 Define a new random variable with this distribution gt Z RandomVariable U PDF Z t t lt 3 0 otherwise Calculate the mean value of the random variab
256. ns sadgccrcunicgeocsacecacsenasaceateennaeyaeietuensaucavounereceas 353 Evaluati EXpPrEsSiONS sirrsreririrrrestfrEEreEri u TEE ATE ga TuE ETEC EEEE OEE EES 359 9 Basio Programimi n sisrerrirseoiirrEens cociros errre EESE CEEA EEEE 369 r TE COMO T ane E TA E EE EA E E S 369 DLF OO a E A EAEE A EEA E E A AEO 370 Conditional Execution 1f Statement cai cceccdansacssecacesinacosantamecgvencceadeanseacens 370 Repetition for Statement ses sescaseuetsanpssteneevonssvaddsocetereumereereserpetenerdnceceore 373 Sr E O eE E EEE E E A ee E AES 380 Feat ASCE aerea E E E EEEE 380 Adding and Multiplying Expressions ccccccec ence cece eee eeseeeeneeaenseneenes 381 Selecting Expression Operands eyescdsn cowed saseerdncaieiansan ae eebeaeenaeusere nents 381 Mapping a Command over a Set or List ccc ccc ec ese ec eee eneeneeaeeseneenees 382 Mapping a Binary Command over Two Lists or Vectors ccceeeeeeeeeeees 383 Additional AMTOTMAUOM 2cesenuaceaelodedac se eshoeredsaenwondervordordenrsawentenncenererteneren 383 ee PI OCCURS aE E EE TEE EE EEE E EE EEE E E E E Ea 383 Defining and Running Simple Procedures ccccsccss0sscatisassdesesdaianssindeacsdusseoses 383 Procedures with Inputs icc can pes ence aang st cnss ernenouaoe an eetunauateeseeskon ee 384 Procedur Remit VAIS gaccecicccuiecatagtecesonsenstenumie ee atesa se cas tectaar etic eae 385 Displaying Procedure Definitions args tees ecshe se seven worsacenereionpuenatennees
257. nsert a task 1 From the Tools menu select Options 2 Click the Display tab 3 In the Show task variables on insert drop down list select Always 4 Click Apply to Session or Apply Globally as necessary 1 3 Point and Click Interaction 43 Updating Parameters and Executing the Commands In inserted Task Templates parameters are marked as placeholders in purple text or spe cified using sliders or other embedded components 1 Specify values for the parameters in placeholders or using graphical interface components You can move to the next placeholder by pressing Tab 2 Execute all commands in the task by e Placing the cursor in the first task command and then pressing Enter repeatedly to execute each command e Selecting all the template commands and then clicking the execute toolbar icon f 3 If the template contains a button that computes the result click it For more information on task templates refer to the tasks help page Exploration Assistant The Exploration Assistant allows you to interactively make parameter changes to expressions and view the result The assistant can be used with almost any Maple expression or command that has at least one variable or parameter To launch the Exploration Assistant 1 Enter an expression or command 2 Right click Control click Macintosh the expression or command From the context menu select Explore 3 The Explore parameter selection dialog appears
258. nt See Units help index for details First select a dimension from the drop down box Then select the units to convert from and to Click the Perform Urat Conversion button The Convert Back button converts in the opposite direction Convert Result 2 831684859 cubic Feet ft 3 cubic meters m3 Dimension volume Perform Unit Conversion Convert Back Figure 4 6 Units Calculator Assistant To perform a conversion 1 In the Convert text field enter the numeric value to convert 2 In the Dimension drop down list select the dimensions of the unit 3 In the From and To drop down lists select the original unit and the unit to which to convert 4 Click Perform Unit Conversion The same conversion can be done with the convert units command gt convert 1 0 units lbffi radius N m radius 1 355817948 Using the Units Calculator you can convert temperatures and temperature changes e To perform a temperature conversion in the Dimension drop down list select temper ature absolute To perform a temperature change conversion in the Dimension drop down list select temperature relative To convert temperature changes the Units Calculator uses the convert units command For example an increase of 32 degrees Fahrenheit corresponds to an increase of almost 18 degrees Celsius 4 5 Units Scientific Constants and Uncertainty 131 gt convert 32 0 units degF degC
259. nuals Resources and more List of Packages Table 3 2 Top Packages CodeGeneration The Code Generation package is a collection of commands and subpack ages that enable the translation of Maple code to other programming lan guages such as C Fortran MATLAB Visual Basic and Java Linear Algebra The Linear Algebra package contains commands to construct and manip ulate Matrices and Vectors and solve linear algebra problems LinearAl gebra routines operate on three principal data structures Matrices Vectors and scalars Optimization The Optimization package is a collection of commands for numerically solving optimization problems which involve finding the minimum or maximum of an objective function possibly subject to constraints Physics The Physics package implements computational representations and related operations for most of the objects used in mathematical physics computa tions RealDomain The Real Domain package provides an environment in which Maple as sumes that the basic underlying number system is the field of real numbers instead of the complex number field ScientificConstants The Scientific Constants package provides access to the values of various physical constants for example the velocity of light and the atomic weight of sodium This package provides the units for each of the constant values allowing for greater understanding of an equation The package also provides units matching for error c
260. nuals from within Maple including the Maple Programming Guide and this manual You can execute examples copy content into other documents and search the contents using the Maple Help System e From the Help menu select Manuals Resources and more and then Manuals Task Templates Set of commands with placeholders that you can use to quickly perform a task For details see Task Templates page 41 e From the Tools menu select Tasks and then Browse Maple Tour and Quick Resources Maple Tour The Maple Tour consists of interactive sessions on several of the following topics Ten Minute Tour Numeric and Symbolic Computations Matrix Computations Differential Equations Statistics Programming and Code Generation Units and Tolerances and Edu cation Assessment with Maple T A e From the Help menu select Take a Tour of Maple Quick Help and Quick References The Quick Help dialog is a list of key commands and concepts e From the Help menu select Quick Help Alternatively press F1 For additional inform ation click an item in the Quick Help The Quick Reference is a table of commands and information for new users that opens in a new window It contains hyperlinks to help pages for more information e From the Help menu select Quick Reference Alternatively press Ctrl F2 Command F2 for Macintosh 60 1 Getting Started Web Site Resources Welcome Center A Maple web site offering all of Maplesoft s key use
261. numbers all equation labels and updates the label references e If you change the equation label format see Label Numbering Schemes page 99 Maple automatically updates all equation labels and label references For information on assigning to using and unassigning names see Names page 94 For more information on equation labels refer to the equationlabel help page The following chapters describe how to use Maple to perform tasks such as solving equations producing plots and animations and creating mathematical documents The chapters were created using Worksheet mode Except where noted all features are available in both Worksheet mode and Document mode 4 Basic Computations This chapter discusses key concepts related to performing basic computations with Maple It discusses important features that are relevant to all Maple users After learning about these concepts you will learn how to use Maple to solve problems in specific mathematical disciplines in the following chapter 4 1 In This Chapter Symbolic and Numeric Computation page 102 Exact Computations An overview of exact and floating point computa Floating Point Computations tion Converting Exact Quantities to Floating Point Values Sources of Error Integer Operations page 106 How to perform Important Integer Commands integer computations Nonsitaces 0 Nuniber Finite Rings and Fields Gaussian Integers Solving Equations page 111 How t
262. nverting Exact Quantities to Floating Point Values ccccceeeee eee ees 104 POU PCCS ON EOT aoee a sacneciss eetonas EAEE TAN E S 105 Os AIC CG OperalonS acececensrs dour ensoaansenour anes coven neon eae eonnouseaensesnseoueousers 106 Non Base 10 Numbers and Other Number Systems c cccceceeceeeeeees 108 A OWI EQUATOR sei caceorentasneradeseceed Gacees nese panp ons reaieaewasicstucieunieetae lt ies ee 111 Contents v Solving Equations and Inequations cccccccc ccc ec ence eee eee eee eeeeaeeseneenees 112 Oi tlelaey oe Cl Ac elec 8 fal ee ee en ee a ee ence ener nets 121 4 5 Units Scientific Constants and Uncertainty ccsscecsscccciscscenroecatustastencsetatetsees 128 C e ce oe eee E E E EEE eee pai nee EEE 128 Scientific Constants and Element Properties sccsssc icaceavasviesiecentastaesencsecsnonss 134 Uncertainty Propagation gases cuscciieesperenaarensdstievarvateurersneiieesiouderedeutseaceniee 139 46 Restricting the DOMAIN ace ic ease seecicerats Cac geaarstotee gerbe ners ep En E EEEE NENE 143 Rer NUMO r DOAI csar E E ERE 143 Assumptions on Variables nosonnssnnnssenesssrresserssssresseressrereseressseressne 144 gt Mathematical Problem Solving ssrsretirrrnra tirrr n nint rti ANENE E EAEE TATEN 149 sT E aO EE E E E EE E S E EE T 149 T ADIS a E E E AEE AE E E E EEE EE EES 150 FPolynon nala Aa DIT sieer AANEEN EEE EE aeenteeeat 150 E e e E S N EE EE E A TE E T A E TTE 159 Creatin
263. nything Storage rectangular Order Fortran order To edit or view a large matrix or vector double click the placeholder This launches the Matrix Browser See Figure 5 5 Insert Figure 5 5 Matrix Browser 164 5 Mathematical Problem Solving To modify the entries using the Matrix Browser 1 Select the Table tab 2 Double click an entry and then edit its value Press Enter 3 Repeat for each entry to edit 4 When you have finished updating entries click Done You can view the matrix or vector as a table or as an image which can be inserted into the document For more information refer to the MatrixBrowser help page To set the maximum dimension of matrices and vectors displayed inline e Use the interface command with the rtablesize option For example interface rtablesize 15 For more information refer to the interface help page Creating Matrices and Vectors with Specific Properties By default matrices and vectors can store any values To increase the efficiency of linear algebra computations create matrices and vectors with properties You must specify the properties for example the matrix shape or data type when defining the object The Matrix palette Figure 5 2 supports several properties To specify the matrix type e Use the Shape and Type drop down lists To specify the data type e Use the Data type drop down list For example define a diagonal matrix with small integer co
264. o access an example worksheet 1 In the worksheet enter index examples The Example Worksheet Index opens 2 Expand the Calculus topic 3 Click the examples Calculus1IntApps link The Calculus1 Applications of Integration worksheet opens See Figure 5 17 4 Expand the Volume of Revolution topic 5 Examine and execute the examples Calculus 1 Applications of Integration The Student Calculus package contains four routines that can be used to both work with and visualize the concepts of function averages arc lengths and volumes and surfaces of revolution This worksheet demonstrates this functionality For further information about any command in the Calculus package see the corresponding help page For a general overview see Calculus Getting Started pees any command in the package can be referred to using the long form for exarnple Student Calculus1 DerivativePlot tt is easter and often clearer to load the package and then use the short form command names restart gt with Student Calculus The following sections show how the routines work In some cases exarnples show to use these visualization routines in conjunction with the single stepping Calculus routines gt Function Average b Volume of Revolution Arc Length gt Surface of Revolution EG Visualization Previous Integration Figure 5 17 Example Worksheet Check for Other Ready To Use Resources Application Center The Ma
265. o manipulate or solve the problem You can also create document blocks in Worksheet mode to perform the same function Document blocks are typically collapsed to hide the Maple code but these regions can also be expanded to reveal this code 52 1 Getting Started To create a document block From the Format menu select Create Document Block If text or math in one or more execution groups 1s selected then a document block 1s created that contains those execution groups If not anew document block is created after the current execution group For more information see the next example Document block regions are identified using markers that are located in a vertical bar along the left pane of the document See Figure 1 15 In addition to document block boundaries these markers icons indicate the presence of hidden attributes in the document such as annotations bookmarks and numeric formatting To activate markers From the View menu select Markers See Figure 1 15 wil aw L C 20 Math 7 Times Mew Roman Figure 1 15 Document Block Markers To view code in a document block 1 Place the cursor in a document block to be expanded 2 From the View menu select Expand Document Block C 20 Math Times Mew Roman de E U Sr h e 10x 21 x 10x421 1 2 Figure 1 16 Expanded Document Block With the Document Block expanded you can see the Maple command that was used to perform
266. o solve Equations and Inequations standard mathematical equations Ordinary Differential Equations Partial Differential Equations Integer Equations Integer Equations in a Finite Field Linear Systems Recurrence Relations 101 102 4 Basic Computations Units Scientific Constants and Uncer Units tainty page 128 How to construct and compute with expressions that have units scientific con stants or uncertainty e Applying Units to an Expression e Performing Computations with Units e Conversions e Changing the Current System of Units e Extensibility Scientific Constants e Scientific Constants e Element and Isotope Properties e Value Units and Uncertainty e Performing Computations e Modification and Extensibility Uncertainty Propagation e Quantities with Uncertainty e Performing Computations with Quantities with Uncertainty Restricting the Domain page 143 How to restrict Real Number Domain the domain for computations e Assumptions on Variables 4 2 Symbolic and Numeric Computation Symbolic computation 1s the mathematical manipulation of expressions involving symbolic or abstract quantities such as variables functions and operators and exact numbers such as integers rationals 7 and e The goal of such manipulations may be to transform an expression to a simpler form or to relate the expression to other better understood formulas Numeric computation is the manipulation
267. ock and use context menus again 4 From the Format menu select Create Document Block 5 To copy the expression 2x 9 0 highlight only this expression from the previous result Press and hold the Ctrl key and drag the expression to the new document block region Result a to left isolate fi Fe Ge Se oe a A 2 lt mowe to left isolate for x My sx f 3x i Bee iol gt i _ mowe to left isolate for x M eE PS 35K A a 2x l 2x 9 1 1 Introduction to Maple 17 To plot the expression 6 Right click the equation and select Left hand Side Input A4x 4 9 0 Copy as MathML Paste Ctrl V Evaluate Evaluate and Display Inline Ctrl Explore pply a Command Differentiate Evaluate at a Point Expand Integrate I Left hand Side Manipulate nin Move to Right Negate Relation Right hand Side Sequence I Simplify b Solve p Test Relation Conversions b Integral Transforms b Plots b 2 D Math d Result 1x 9 0 left hand side 47 9 7 Right click the expression and select Plots 2 D Plot 18 1 Getting Started Copy as MathML Faste Explore pply a Command Assign to a Mame Coefficients Collect Differentiate Evaluate at a Point Factor Integrate Lirnik SeQuence Series Simplify Solve Complex Maps Constructions Conversions Integer Functions Integral Transforms Language Conversions Optimization
268. of operators and other expressions see Distinction between Functional Operators and Other Expressions page 344 To find the derivative of a functional operator e Use the D operator The D operator returns a functional operator For example find the derivative of an operator that represents the mathematical function F x xcos x First define the operator F 1 In the Expression palette click the single variable function definition item 54 gt 2 Enter placeholder values e To move to the next placeholder press the Tab key Note If pressing the Tab key inserts a tab click the Tab icon in the toolbar gt F x gt xcos x Now define the operator G that maps x to the derivative of xcos x gt G D F T F and G evaluated at return the expected values 180 5 Mathematical Problem Solving For more information on the D operator refer to the D help page For a comparison of the diff command and D operator refer to the diffVersusD help page Directional Derivative To compute and plot a directional derivative use the Directional Derivative Tutor The tutor computes a floating point value for the directional derivative To launch the tutor From the Tools menu select Tutors Calculus Multi Variable and then Directional Derivatives Maple launches the Directional Derivative Tutor See Figure 5 7 5 4 Calculus 181 Multivariate Calculus Directional Derivative
269. om the context menu select Show Palette and then Components For more information see Palettes page 22 You can embed the following items e Button Toggle Button e Combo Box Check Box List Box Radio Button e Text Area Label e Slider Plot Mathematical Expression e Dial Meter Rotary Gauge Volume Gauge W Components Button Toggle Button Combo Box _ Check Box Radio Button Text Area Figure 7 21 Components Palette e 331 Task Template with Embedded Components In your document you can add components that have already been configured to work to gether by using a task template Here we use the template For details on how to create and modify components see Creating Embedded Components page 393 To insert the task template from the Tools menu select Tasks Browse In the table of contents expand Document Templates and select Interactive Application Click Insert Minimal Content The following is inserted into your document Title author Explanatory text describing the application Title use the Dials to set parameters parameter1 parameter2 Plot of yan Figure 7 22 Interactive Application Task Template This configuration of components plots a linear function with slope and y intercept given respectively by the two dials parameter2 and parameter and displays the function talc on a gauge For details on how these components work together see Embedded parame
270. omic weight of sodium The ScientificConstants package also provides the units for the constant values allowing for greater understanding of the equation as well as unit matching for error checking of the solution The quantities available in the ScientificConstants package are divided into two distinct categories 4 5 Units Scientific Constants and Uncertainty 135 e Physical constants e Chemical element and isotope properties Scientific Constants List of Scientific Constants You have access to scientific constants important in engineering physics chemistry and other fields Table 4 5 lists some of the supported constants For a complete list of scientific constants refer to the ScientificConstants PhysicalConstants help page Table 4 5 Scientific Constants elementary charge S You can specify a constant using either its name or symbol Accessing Constant Definition The GetConstant command in the ScientificConstants package returns the complete definition of a constant To view the definition of the Newtonian gravitational constant specify the symbol G or its name in a call to the GetConstant command gt with ScientificConstants gt GetConstant G se n nine nell s13 oec Newtonian constant of gravitation symbol G value 6 67310 uncertainty 1 0 10 units Jj mi ia ke s For information on accessing a constant s value units or uncertainty see Value Units and Uncertainty pa
271. ommand solves quadratic programs gt Optimization LPSolve c A b maximize assume nonnegative I 20 Vector column ae a Data Type float 43 2673034492019 Storage rectangular Order Fortran order This example uses a random data set to demonstrate the problem You could also read data from an external file as Matrices and use that data For details and an example see Reading from Files page 414 Note For information on creating matrices and vectors including how to use the Matrix palette to easily create matrices see Linear Algebra page 159 192 e 5 Mathematical Problem Solving For additional information on performing efficient computations refer to the Optimiza tion Computation help page MPS X File Support To import linear programs from a standard MPS X data file use the ImportMPS command Optimization Package Commands Each Optimization package command solves the problem using a different optimization method These are described in Table 5 9 along with the general input form for each com mand Table 5 9 Optimization Package Commands LPSolve Solve a linear program LP which involves computing the minimum or maximum of a linear objective function subject to linear constraints input is in equation or Matrix form LSSolve Solve a least squares LS problem which involves computing the minim um of a real valued objective function having the form i rule x fr j shar
272. omputations and create customizable zoomable graphs This is available on Microsoft Windows only This manual describes how to use the Standard interface As mentioned the Standard inter face offers two modes Document mode and Worksheet mode Using either mode you can xiii xiv Preface create high quality interactive mathematical documents Each mode offers the same features and functionality the only difference is the default input region of each mode Shortcut Keys by Platform This manual will frequently refer to context menus and command completion when entering expressions The keyboard keys used to invoke these features differ based on your operating system This manual will only refer to the keyboard keys needed for a Windows operating system The shortcut keys for your operating system can be viewed from the Help menu Help Manuals Resources and more Shortcut Keys Context Menus Right click Windows and UNIX e Control click Macintosh That is place the mouse over the input or output region and press the right button on the mouse or press and hold the Control key and click the mouse key for Macintosh For more information on Context Menus see Context Menus page 39 Command Completion e Esc Macintosh Windows and UNIX e Ctrl Space Windows e Ctrl Shift Space UNIX Begin entering a command in a Maple document Press the Esc key Alternatively use the platform specific
273. ompute eigenvalues eigenvectors and singular values e Compute the dimension or rank e Convert to the Jordan form or other forms e Perform Cholesky decomposition and other decompositions 172 e 5 Mathematical Problem Solving For example compute the infinity norm of a matrix See Figure 5 6 18735 6985 349723 234987 9859 459 798124 14089 Copy as MathML Paste Cret Explore Apply a Command Approximate 45sign to a Mame Browse Export 45 Map Command Onto Morm b 1 Select Elements b Euclidean infinity Conversions k Frobenius Curve Fitting Eigenvalues etc In Place Options Language Conversions Map Integer Functions Onto Plots Queries Solvers and Forms i i F F Standard Operations 2 D Math il Figure 5 6 Computing the Infinity Norm of a Matrix In Document mode Maple inserts a right arrow and the name of the computation performed followed by the norm Le fS5 6985 S49 25 254987 infinity norre yp 9659459 9sl 24 14089 80798359990 10 Vector operations available in the context menu include the following e Compute the dimension e Compute the norm 1 Euclidean and infinity e Compute the transpose 5 3 Linear Algebra 173 e Select an element For more information on context menus see Context Menus page 70 for Document mode or Context Menus page 89 for Worksheet mode LinearAlgebra Package Commands The LinearAlgebra package contains commands that construct and
274. on of that quantity use the evalf command or the Approximate context menu operation see Approximating the Value of an Expression page 71 gt evalf z evalf sin 3 evalf gt 3 141592654 0 141 1200081 1 833333333 By default Maple computes such approximations using 10 digit arithmetic You can modify this in one of two ways e Locally you can pass the precision as an index to the evalf call 4 2 Symbolic and Numeric Computation 105 gt evalf 20 exp 2 evalf 3 7 3890560989306502272 1 354117939 e Globally you can set the value of the Digits environment variable gt Digits 25 gt evaif tan a 1 732050807568877293527446 For more information see the evalf and Digits help pages Note When appropriate Maple performs floating point computations directly using your computer s underlying hardware Sources of Error By its nature floating point computation normally involves some error Controlling the effect of this error is the subject of active research in Numerical Analysis Some sources of error are I l e An exact quantity may not be exactly representable in decimal form and 7 are examples e Small errors can accumulate after many arithmetic operations e Subtraction of nearly equal quantities can result in essentially no useful information For example consider the computation x sin x for x 0 gt x sin x 00001 No correct digits remain If however
275. op level command enter its name followed by parentheses containing any parameters This is referred to as a calling sequence for the command command arguments Note In 1 D Math input include a semicolon or colon at the end of the calling sequence For example to differentiate an expression use the diff command The required parameters are the expression to differentiate which must be specified first and the independent variable gt diff tan x sin x x 4 1 tan x sin x tan x cos x For a complete list of functions commands that implement mathematical functions such as Bessell and AiryAi available in the library refer to the initialfunctions help page BesselI 0 1 1 AiryAi 2 2 47 53037086 For detailed information on the properties of a function use the FunctionAdvisor command gt FunctionAdvisor definition Bessell a ri z hypergeom 1 a 17 Bessell a z with no restrictions on a z r 1 a 2 For detailed information on how to use a function in Maple refer to its help page For example gt Bessel Note In 1 D and 2 D Math input when accessing a help page using you do not need to include a trailing semicolon or colon 84 3 Worksheet Mode Top Commands Here are a few of the most frequently used Maple commands A complete list of top level commands is available at Help Manuals Resources and more List of
276. or ayaz Numerically Solve Title Complex Maps b i p p Numerically Solve From point ci a Constructions Obtain Solutions For Manipulator Conversions b l Solve Export b Integer Functions b i hs Solve Cescplicik Integral Transforms d Solve general solution Language Conversions b s1ag Solve For Variable b Optimization d Plots b Sorts d Units b Figure 1 8 Right click the expression to seea Figure 1 9 Right click the plot to see a menu menu of applicable operations of plot options 1 3 Point and Click Interaction 41 Task Templates Task templates help you perform specific tasks in Maple such as e performing a mathematical computation such as solving an equation symbolically or numerically or determining the Taylor approximation of a function of one variable e constructing a Maple object such as a function e creating a document such as an application Each task contains a description along with a collection of content that you can insert directly into your document Content consists of 2 D mathematics commands embedded components for example buttons and plots You specify the parameters of your problem and then execute the commands in the document See Figure 1 10 for an example of a Task Template g Browse Tasks Jo File view pe n F Overview D 4 9 Algebra Gy Calculus Differential Calculus Integral Copy Task to Clipboard Insert Default Content Insert Minim
277. ough the Paragraph Style dialog Headers and Footers You can add headers and footers to your document that will appear at the top and bottom of each page when you print the document To add or edit headers and footers From the View menu select Header Footer The Header Footer dialog appears See Figure 7 8 Header and Footer Insert Date Insert Page Insert Number of Pages Insert Picture Insert File Name Left Center Right Figure 7 8 Header and Footer Dialog Custom Header The available elements include the current date page number number of pages an image the filename or any plain text These elements can be placed in the left or right corner or the center of the page 298 e 7 Creating Mathematical Documents You can choose one of the predefined header or footer styles in the Predefined Header and Footer tab or create your own by clicking the Custom Header or Custom Footer tab For more information on header and footer options refer to the headerfooter help page Show or Hide Worksheet Content You can hide document elements of a specific type so that they are not visible This does not delete them but hides them from view Hidden elements are not printed or exported but they can be copied and pasted In a document use the Show Contents dialog to hide all spreadsheets input output or graphics plus markers for section boundaries execution group boundaries hidden tab
278. oximating the Value of a Fraction at 10 digits R 3 6666666667 You can replace the inserted right arrow with text or mathematical content To replace the right arrow gt 1 Select the arrow and text Press Delete 2 Enter the replacement text or mathematical content Note To replace the right arrow with text you must first press F5 to switch to Text mode For example you can replace the arrow with the text is approximately equal to or the symbol 3 is approximately equal to 0 6666666667 2 66660666667 2 6 Performing Computations 73 Solving an Equation You can find an exact symbolic solution or an approximate numeric solution of an equation For more information on symbolic and numeric computations see Symbolic and Numeric Computation page 102 To solve an equation 1 Enter an equation 2 Display the context menu See Figure 2 3 3 From the context menu select Solve or Numerically Solve in the Solve menu item 74 2 Document Mode ual om a Copy as MathML Paste Ctrl Evaluate Evaluate and Display Inline Ctrlt Explore 4oply a Command Approximate H Combine Complete Square Cross Multiply Differentiate Evaluate at a Point Expand Integrate Left hand Side Manipulate Equation Map Command Onto Move to Left Move to Right Negate Relation Right hand Side Sequence Simplify p Solve Isolate Expression For b Test Relation Numerically Solve
279. p down menu 4 To insert a symbol click the displayed symbol 1 2 Entering Expressions 29 W Handwriting o fe al Figure 1 3 Handwriting Palette For more information refer to the handwritingpalette help page Symbol Names Each symbol has a name and some have aliases By entering its name or an alias in Math mode you can insert the symbol Note If you hover the mouse pointer over a palette item a tooltip displays the symbol s name To enter a symbol quickly you can enter the first few characters of its name and then press the completion shortcut key Esc see Shortcut Keys by Platform page xiv Symbol completion works in the same way as command completion see Command Comple tion page 48 e Ifaunique symbol name matches the characters entered Maple inserts the corresponding symbol e If multiple symbol names match the characters entered Maple displays the completion list which lists all matches including commands To select an item click its name or symbol Example 4 Square Root To find the square root of 603729 30 1 Getting Started 2 Press the symbol completion shortcut key grr Esc Maple displays a pop up list of exact matches 3 In the completion list select Ea Mapie inserts the symbol with the x placeholder selected 4 Enter 603729 603729 Example 5 Complex Numbers When you simply type the letter i in Math mode it is in italics This lett
280. ple 211 Technical Support access 61 temperature conversion 130 Text Area component 391 text field embedding 329 Text mode 20 text regions 93 tilde 116 144 363 to clause 374 excluding 374 Toggle Button component 391 Tolerances package 139 toolboxes Global Optimization 188 Tools menu assistants 33 Assistants and Tutors 91 Tasks 91 topic search 56 Torsion command 187 total degree 153 transparency of 3 D plots 267 transpose matrices and vectors 171 true 370 Tutorials 58 tutorials help system 56 Tutors 198 199 201 Derivatives 199 Differentiation Methods 200 440 Index Directional Derivative 180 Gradient 201 tutors accessing 38 Linear System Solving 76 using 37 type command 349 types 144 348 converting 356 series 183 testing 348 subexpressions 349 typesetting rule assistant 299 U unapply command 120 unassign command 95 unassigning names 95 368 uncertainty 139 quantities with 139 underline format 284 unevaluation quotes 96 366 union of sets 339 Unit Converter Assistant 356 Units package description 87 units 75 128 356 adding to expressions 75 applying to expression 131 computing with 132 context 129 converting between 129 environment 132 evaluating with 75 in 1 D Math 132 inserting 131 overview 128 prefixes 132 system of controlling 133 systems of 129 Units Calculator 129 Units Calculator Assistant 37 Un
281. ple Application Center contains free user contributed applications related to math ematics education science engineering computer science statistics and data analysis finance communications graphics and more 212 e 5 Mathematical Problem Solving To access a free application for volume of revolution 1 Go to the Maplesoft web site http www maplesoft com 2 In the menu of the main web page click Community and then Application Center 3 In the Application Search section enter Calculus 2 in the Keyword or phrase field Application Search Calculus 2 Any Application Typel Search CA Advanced Search 4 Click Search 5 From the search results page under Displaying applications click the Click here link 6 From the list of archived applications select Calculus I Complete Set of Lessons 7 Click on the Download Maple Document link TPE Toolkit Download Maple Document EQ view HTML Version Tella Colleague about this Application eo Contact the Author Contribute Your ork en Evaluate Maple 8 Log in as a guest or Maplesoft Member 9 Download the zip file 10 Extract the L2 volumeRevolution mws file 11 Execute the worksheet and examine the results 5 8 Clickable Math Examples 213 5 8 Clickable Math Examples Maple has incorporated several features that eliminate the learning curve for new users With drag and drop functionality context menus built in tutors co
282. ple lists the command s calling sequence parameters and a description with examples of the command at the end of the page Some help pages also contain hyper links to related help pages and hyperlinks to dictionary definitions Hyperlinks to help pages display in green while hyperlinks to dictionary definitions display in dark red 56 1 Getting Started Using the Help Navigator The Help Navigator contains a field for topic or text based searches The Table of Contents tab provides a structured list of all topics in the help system To search the help system 1 In the left pane enter a string in the search field 2 By default a topic search is performed To perform a text search select the Text radio button 3 Enter the term and click Search e Topic searches reveal a list of matching topics sorted by the precision of the match e Text searches reveal a list of topics based on keyword frequency e You can search all of the help system or specific Resources such as Help Pages Tasks Tutorials and Manuals by selecting the Resources drop down menu Search results are displayed as a list in the Search Results tab of the left pane Click the Table of Contents tab to view a structured list of all topics in the help system To display potential matches in the right pane click a topic preceded by an icon Table 1 9 describes the different icons Table 1 9 Help Page Icons A folder icon in the Table of Contents tab indicates
283. plication 1 In the Type drop down list select Maplet 2 In the Target field enter the local path to a file with the maplet extension Optionally click Browse to locate the file If the Maplet application exists clicking the link launches the Maplet application If the Maplet application contains syntax errors then error messages are displayed in a pop up window When linking to a custom Maplet application the path is absolute When sharing documents that contain links to Maplet applications ensure that target Maplet applications are in the same directory 7 6 Hyperlinks 327 3 Click OK Note To link to a Maplet application available on a MapleNet Web page use the URL hyperlink type to link to the Web page For information on MapleNet see Embedded Components and Maplets page 389 Example For this example link the text horizontal range to the dictionary page for domain As in dicated in the section for Linking to a Dictionary Topic select Dictionary Topic in the Type drop down list and then enter Definition domain in the Target field Links to dictionary topics appear underlined and in red Result plot create a two dimensional plot Calling Sequence plottf x plot f x x0 x1 plot w1 v2 Parameters f BEpression in independent variable x F independent variable 0 x1 le and right endpomts of horizontal range il 2 x coordmates and y coordinates Bookmarks Us
284. r lt and using the lt gt lt and gt operators The evalb command uses a three valued logic system The return values are true false and FAIL If evaluation is not possible an unevaluated expression is returned 8 3 Working with Maple Expressions 363 gt evalb x x rue gt evalb x y false gt evalb 3 21 lt 2 4 3 7 FAIL Important The evalb command does not perform arithmetic for inequalities involving lt S gt or anddoes not simplify expressions Ensure that you perform these operations before using the evalb command gt evalb R x lt R x 1 ne lt 1 FRE gt evalb R x R x 1 lt 0 rue Applying an Operation or Function to All Elements in a List Set Table Array Matrix or Vector You can use the tilde character to apply an operation or function to all of the elements in a list set table Array Matrix or Vector In the following example each element in the Matrix M is multiplied by 2 by adding a tilde character after the multiplication operator 364 8 Maple Expressions 23 gt M 43536 789 L23 M 456 8 6 789 gt M 2 2 4 6 8 amp 8 10 12 8 7 14 16 18 In the following example the function sin is applied to each element in the Matrix M gt sin M sin 1 sin 2 sin 3 sin 4 sin 5 sin 6 8 8 sin 7 sin 8 sin 9 The tilde character can also be used to apply a function to multiple data sets for example gt
285. r Oper ations page 106 5 2 Algebra 151 Table 5 1 Polynomial Arithmetic ea Eumple SSS eer Multiplication x 3 3 ip Sy 3x 5x 2 r P n Ee t 5x Exponentiation Division Quotient and Remainder You can specify multiplication explicitly by entering which displays in 2 D Mathas In 2 D Math you can also implicitly multiply by placing a space character between two expressions In some cases the space character is optional For example Maple interprets a number followed by a name as an implicit multiplication In 2 D Math exponents display as superscripts To expand a polynomial use the expand command 152 e 5 Mathematical Problem Solving gt expand 3 3x 5 x 2 9x 14x 2 If you need to determine whether one polynomial divides another but do not need the quotient use the divide command The divide command tests for exact polynomial division es i 9 39 T 69 2 gt divide lt 4 Fry e y 13 x2 13 x 13 y Heyes r frue Important You must insert a space character or a multiplication operator between adjacent variables names Otherwise they are interpreted as a single variable For example x does not divide the single variable xy gt divide xy x false But x divides the product of x and y gt divide xy x divide x y x true true For information on polynomial arithmetic over finite rings and fields refer to the mod help pag
286. r resources in one central location In the Welcome Center you can view sample applications participate in user forums access exclusive premium content and listen to podcasts You can also access our support services view training videos download user manuals and more http www maplesoft com welcome Student Help Center The Student Help Center offers a Maple student forum online math Oracles training videos and a math homework resource guide http www maplesoft com studentcenter Teacher Resource Center The Teacher Resource Center is designed to ensure you get the most out of your Maple teaching experience It provides sample applications course material training videos white papers e books podcasts and tips http www maplesoft com teachercenter Application Center Maple web site resource for free applications related to mathematics education science engineering computer science statistics and data analysis finance communications and graphics Many applications are available in translations French Spanish and German You can also search for Education and Research PowerTools which provide free course curricula and are available as add on Maple packages and courses PowerTools are developed by experts in their fields to help users configure Maple for research 1n specific application areas http www maplesoft com applications Training Maplesoft offers a comprehensive set of complementary
287. raded Note This feature can be used to create questions for Maple T A an online automated testing and assessment system For details about Maple T A see Maple T A page 419 Creating a Question To create a question 1 Open the Task browser Tools Tasks Browser 2 From the Maple T A folder select the appropriate question type 3 Insert the question template into a document 4 Enter the question content as described in the template 5 Repeat steps to 4 for each question to add to the document Viewing Questions in Maple To view and test your questions in Maple e From the View menu select Assignment This view displays all of the questions in your assignment with access to hints plotting and grading After answering your questions you can test the grading function by clicking the Grade button A Maplet dialog is displayed indicating if the question was answered correctly If hints were provided in the question these are also displayed 7 10 Worksheet Compatibility 335 Saving Test Content When you save a document with test content the authoring and assignment modes determine what the user sees when opening your document e If you save the document in authoring mode task template contents visible the user sees this content when opening the document e If you save the document in assignment mode the user sees only the assignment layout In both cases the View Assignment menu is acc
288. rand of expression to variable 4 Repeat steps 2 and 3 for each operand in expression If there are no more operands exit the loop This is the loop bound test Example The following loop returns a floating point approximation to the sin function at the angles measured in degree in the list L gt L 23 4 87 2 43 0 99 7 gt for iin L do evalf sin i Pi 180 end do 0 3971478907 0 9988061374 0 6819983602 0 9857034690 while Loop The while loop repeats a statement sequence until a boolean expression does not hold 9 2 Flow Control 377 Syntax The while loop has the following syntax while conditional expression do SLalenentl seqcence end do A while loops repeats until its boolean expression conditional_expression evaluates to false or FAIL For more information on boolean expressions see Conditional Execution if Statement page 370 Example The following loop computes the digits of 872 349 in base 7 in order of increasing signi ficance gt x 872349 378 9 Basic Programming gt while x gt 0 do irem x 7 x iquo x 7 end do x 124621 b2 To perform such conversions efficiently use the convert base command 9 2 Flow Control 379 gt convert 872349 base 7 2 0 2 2 6 2 0 1 For information on non base 10 numbers see Non Base 10 Numbers page 108 General Loop Statements You can include a while statement in a for from or for
289. ration gt xsin ax xsin ax 5 4 gt int 5 4 x sin ax xcos ax a 5 5 ar For a definite integration set the variable of integration equal to the interval of integration T gt in 6 4 1 0 a 5 6 Numeric Integration To perform numeric integration e Use the evalf Int arguments calling sequence Important Use the inert Int command not the int command For more information refer to the int help page In addition to the arguments accepted by the int command you can include optional argu ments such as method which specifies the numeric integration method 186 5 Mathematical Problem Solving gt evalf mf x 0 2 metho _Dexp x 1 626378399 Note To enter an underscore character _ in 2 D Math enter _ For information on the evalf command see Numerical Approximation page 361 For information on numeric integration including iterated integration and controlling the algorithm refer to the evalf Int help page Differential Equations Maple has a powerful set of solvers for ordinary differential equations ODEs and partial differential equations PDEs and systems of ODEs and PDEs For information on solving ODEs and PDEs see Other Specialized Solvers page 121 Calculus Packages In addition to top level calculus commands Maple contains calculus packages VectorCalculus Package The VectorCalculus package contains commands that perform multivari
290. re Languages Accessing External Products from Maple Accessing Maple from External Products Sharing and Storing Maple Worksheet Content with the MapleCloud 11 2 Writing to Files Maple supports file formats in addition to the standard mw file format After using Maple to perform a computation you can save the results to a file for later processing with Maple or another program Saving Data to a File If the result of a Maple calculation is a long list or a large array of numbers you can convert it to Matrix form and write the numbers to a file using the ExportMatrix command This command writes columns of numerical data to a file allowing you to import the numbers into another program To convert a list or a list of lists to a Matrix use the Matrix construct or For more information refer to the Matrix help page 411 412 e 11 Input Output and Interacting with Other Products 98 76 4 29 38 77 72 27 44 gt LS S 37 2 6 M of 21 32 OY 31 33 93 74 99 67 gt ExportMatrix matrixdata txt L If the data is a Vector or any object that can be converted to type Vector use the Ex portVector command To convert lists to Vectors use the Vector constructor For more information refer to the Vector help page gt R 3 3 1415 65 0 R 3 3 1415 65 0 11 1 gt V Vector R 3 y 31415 re a 11 2 gt ExportVector vectordata txt V You can extend these
291. re visible you create a document with better presentation flow Before using document blocks it 1s recommended that you display Markers A vertical bar is displayed along the left pane of the document Icons representing document blocks are displayed in this vertical bar next to associated content To activate Markers e From the View menu select Markers For further details on document blocks see Document Blocks page 51 in Chapter 1 Working with Document Blocks In document mode each time you press Enter a new document block appears Documents consist of a series of document blocks 1 Create a new document block after the last section of the pasted example either by pressing Enter or by selecting from the Format menu Create Document Block 2 Enter text and an expression to evaluate For example enter Plot the expression sin x and its derivative L sin x For detailed instructions on entering this phrase see Example 6 Enter Text and 2 D Math in the Same Line page 31 in Chapter 1 3 Select the expression Control click for Macintosh to display the context menu 4 Click the Evaluate and Display Inline menu item The expression is evaluated 5 Check that the input mode is Text then enter the rest of the sentence in the same plot See Figure 7 10 7 3 Commands in Documents 301 Before Plot the expression sin x and its derivative sin 2 dx de I Copy as MathML Paste Ctrl Evaluate
292. rksheet Mode page 79 The Startup dialog also contains links to items such as various document options help re sources including updates and other introductory help pages and application resources on the Maplesoft web site Subsequent sessions display Tip of the Day information To start a Maple session 1 In the Startup dialog select Blank Document or Blank Worksheet A blank document displays or 1 Close the Startup dialog 2 From the File menu select New and then either Document Mode or Worksheet Mode A blank document displays Every time you open a document Maple displays a Quick Help pop up list of important shortcut keys To invoke Quick Help at any time press the F1 key Entering 2 D Math In Maple the default format for entering mathematical expressions is 2 D Math This results in mathematical expressions that are equivalent to the quality of math found in textbooks Entering 2 D Math in Maple is done using common key strokes or palette items For more information on palettes see Palettes page 22 An example of entering an expression using common key strokes is presented in the following section An example of entering an ex pression using palette items is presented in Example 3 Enter an Expression Using Palettes page 27 6 1 Getting Started Common Operations 5 99 7 9 Entering mathematical expressions such as xX x and x y is natural in 2 D Math To enter a fraction 1 Enter the
293. rmation on verifying and using solutions returned by the solve command see Working with Solutions page 119 Numerically Solving Equations The fsolve command solves equations numerically The behavior of the fsolve command is similar to that of the solve command gt equation2 zcos z 2 gt fsolve equation2 z 23 64662473 4 8 Note You can also numerically solve equations using the context menus See Solving Equations and Inequations page 112 It is recommended that you verify the solutions returned by the fsolve command For details see Working with Solutions page 119 Multiple Equations To solve multiple equations specify them as a set For more inform ation see Creating and Using Data Structures page 337 The fsolve command solves for all unknowns gt feolve In x j l xy e x 3 396618823 y 0 4719962637 Univariate Polynomial Equations In general the fsolve command finds one real solution However for a univariate polynomial equation the fsolve command returns all real roots gt egquation3 3y y F 118 4 Basic Computations gt fsolve equation3 y 0 3365322739 1 940392664 Controlling the Number of Solutions To limit the number of roots returned specify the maxsols option gt fsolve equation3 y maxsols 1 To find additional solutions to a general equation use the avoid option to ignore known solutions gt fsolve equation2 z avoid z 4 8
294. rmatting elements Oui Charater Forma eae 24 Quick Paragraph Formatting page 286 Character and Paragraph Styles page 288 Sections page 295 Headers and Footers page 297 Show or Hide Worksheet Content page 298 Indentation and the Tab Key page 299 Commands in Documents page 300 Document Blocks page 300 Format and display or hide commands in a document Typesetting page 303 Auto Execute page 304 281 282 e 7 Creating Mathematical Documents Tables page 306 Create tables and Creating a table modify their attributes Cai conden Navigating table cells Modifying Structural Layout Modifying Physical Dimensions Modifying Appearance Printing Options Execution Order Tables in the Classic Worksheet Canvas page 318 Sketch an idea in Insert a Canvas the document by inserting a canvas pa rawing Canvas Style Inserting Images Hyperlinks page 323 and Bookmarks Inserting a Hyperlink in the Document Add hyperlinks to various sources Linking to an Email Address Dictionary Topic Help Page Maplet Application Web Page or Document Bookmarks Embedded Components page 329 In Overview of available components sert buttons sliders and more in your document Spell Checking page 332 Verify text How to Use the Spellcheck Utility with the Maple spell checking utility Example using a task template Selecting a Suggestion User Dictionary Creating Graded Assignments pag
295. ry folder of your choice 2 In Maple open the Options dialog Tools Options and select the General tab 3 In the User Dictionary field enter the path and name of the txt file you created or click Browse to select the location and filename 4 To ignore Maple words that are command and function names clear the Use Maple Words in spellchecker check box 5 Click Apply to Session or Apply Globally to save the settings or Cancel to discard 334 7 Creating Mathematical Documents Adding a Word to Your Dictionary When running the spellcheck if the word in the Not Found text box is correct you can add the word to your dictionary 1 Click the Add button If this is the first time you are adding a word the Select User Dictionary dialog opens 2 Enter or select the custom dictionary txt file you created See User Dictionary page 333 3 Click Select The word is automatically added to your custom dictionary file Note Specifications in the Options dialog determine whether this word is recognized in your next Maple session If you set your custom dictionary and clicked Apply to Session then this word will not be recognized in a new Maple session If you set your custom dic tionary and clicked Apply Globally then this new word will be recognized 7 9 Creating Graded Assignments You can use Maple to create graded assignments Question types include multiple choice essay true or false fill in the blanks and Maple g
296. s gt expand y 3 x 1 x y 3 7 9 3 2 J 3 2 y x f y xX y y 2y y JV x y 3x 6x 3x cay 3y gt expand sin x y sin x cos v cos x sin y Combining To combine subexpressions in an expression e Use the combine command The combine command applies transformations that combine terms in sums products and powers into a single term gt combine sin x cos v cos x sin y sin x y Recall that a was previously assigned to represent a two dimensional array see Creating and Using Arrays page 340 gt zombi el e x 3 SS rx x 9 11 13 I5 17 28 The combine command applies only transformations that are valid for all possible values of names in the expression 356 8 Maple Expressions gt combine 41n x ln y 4In x Iny To perform the operation under assumptions on the names use the assuming command For more information about assumptions see Assumptions on Variables page 144 gt combine 4ln x In y assumingx gt 0 y gt 0 4 A in Converting To convert an expression e Use the convert command The convert command converts expressions to a new form type see Expression Types page 348 or in terms of a function For a complete list of conversions refer to the 2 convert help page Convert a measurement in radians to degrees gt convert n degrees 180 degrees To convert measurements that use units use
297. s Units To obtain the units for a ScientificConstants object use the GetUnit command Ib s gt GetUnit G gt GetUnitLiAtomicWeight For information on changing the default system of units for example from SI to foot pound second see Changing the Current System of Units page 133 Value and Units If you are performing computations with units you can access the value and units for a ScientificConstants object by specifying the units option when constructing the object and then evaluating the object gt evalf Constant G units 1068912061 10 lbs gt evalf Element Li 5 atomicmass units 1835022162 107 7b Uncertainty The value of a constant 1s often determined by direct measurement or derived from measured values Hence it has an associated uncertainty To obtain the uncertainty in the value of a ScientificConstants object use the GetError command 4 5 Units Scientific Constants and Uncertainty 139 gt GetError G iT gt GetError LiAtomic Weight 3 321080400 107 Performing Computations You can use constant values in any computation To use constant values with units use a Units environment as described in Performing Computations with Units page 132 For information on computing with quantities that have an uncertainty see the following section Modification and Extensibility You can change the definition of a scientific constant or
298. s 385 logical operators 370 loops 373 general 379 infinite 379 Macintosh command complete 7 context menus 39 manipulate equation 218 map command 382 Maple Application Center 198 Maple library 45 Maple Portal 58 198 Maple Student Help Center 199 MapleCloud 423 MaplePrimes 61 Maplet Builder description 37 launching 403 Maplet authoring 403 Maplets adding hyperlink to 326 authoring 409 Maplet Builder 403 Maplets package 407 launching Maple worksheet 401 Maplet file type 400 Maplets package Display command 407 Elements subpackage 407 Maplet authoring 407 saving Maple worksheet 409 maplet file 409 using 401 markers Index 433 bookmarks 327 displaying 52 for document blocks 300 math dictionary description 58 math educators portal for 198 Math Expression component 390 Math mode 20 shortcuts 7 mathematical functions list 83 mathematics computations 149 teaching and learning 211 matrices 343 arithmetic 169 context menus 171 data type 164 166 defining 159 efficiency 164 filling 166 Hermitian transpose 171 image 164 large 163 multiplication 169 operations 171 random 163 scalar multiplication 170 selecting submatrices 167 shape 164 166 transpose 171 type 164 Matrix Browser 162 163 342 constructor 165 data structure 159 palette 126 159 164 Matrix command 159 max command 108 maximize 188 maximum 108 Mean command 194 434
299. s cece easuee sce econ oensecrusnsuersens 90 SOAS SA E LULOUS eeeceaicdcenncieesauacs don EEEE EEE eaere 9 Launching an Assistant OF Tutot coed jecsaniegesdansienecsnn tau neumeeareesucsnodeeen cesses 9 Dal Wash E eea E E EA E E E AE 9 DO PEK 1 COIONS ouseccace ew aase cocoon vetetounucnme sts AE AE EEE E EA EET 93 Dee INGINCS e a E in qsennsdestotaacsueincbanantun E E E E E E 94 Assignine tO Names senio trates uactccdematoeaconicensaesinereeiorsgeneerstarnnassmetoosere ns 94 Wass rem NAMES ass access cope awcpu EERE EAEE TEREE TERET EIEEE EEE TERETE 95 NAIC AN AUS E E E E EEE E E E E E E E E ETT 96 OMERE a Ee E E E E EE POE EEEE E E 97 Displaying Equation Labels recnasdrcncawcionseccneentedyeanmeercuecessistmenemunorassnnekecdsness 97 Referring to a Previous CSG aioe cc tienes been ce sesued oanesaceanccnteeaespnoaseeaseeteneaees 97 Execution Groups with Multiple Outputs ccc cece ccc ce eee ec eee eneeaeeseesenees 99 Pet N umbenng Schemes s4 ciccugarcerrsepnesecensecn sone aa E EEEE E 99 Features of Equation Labels cegey reser ticsnsacncacunceeccicceceesaseeceiensecsaespeawennens 100 A Bae CONDI AO e E E E E E EE 101 A T AUC E eae nance EE EE EEEE E EEE E E 101 4 2 Symbolic and Numeric Computation ccc cece ccc ec eee ee eee eeseeeaeeneeseeeas 102 BRACE COPA MONS secperrronin eetere e EENE EEEE ERE EEEE E EER NEARE 103 Floating Point Computations cccceccceeseeececeeceeeececcuteaeeseaceseaeeeens 103 Co
300. s errors Maple Text Maple text is marked text that retains the distinction between text Maple input and Maple output Thus you can export a document as Maple text send the text file by email and the recipient can import the Maple text into a Maple session and regenerate the computations in the original document PDF Export a Maple document to a Portable Document Format PDF file so that you can open the file in a reader such as Adobe Acrobat The PDF document is formatted as it would appear when the Maple worksheet is printed using the active printer settings Note Images plots and embedded components may be resized in the PDF file Plain Text Export a Maple document as plain text so that you can open the text file in a word processor Rich Text Format RTF Export a Maple document to a rich text format file so that you can open and edit the file in a word processor Note The generated rtf format is compatible with Microsoft Word and Microsoft WordPad only 418 11 Input Output and Interacting with Other Products Summary of Translation Table 11 1 Summary of Content Translation When Exporting to Different Formats Content HTML LaTeX Maple Maplet Maple Plain Input Applica Text Text tion Text Maintained Maintained Pre Preceded Preceded Main Main Main ceded by by by tained tained tained H 1 D Math Maintained Maintained Main Main Preceded Preceded Static im
301. s multiple units all units are expressed using units from the current system of units gt 132 25 mile 132 25 mi 4 17 By default Maple uses the SI system of units in which length is measured in meters and time is measured in seconds 4 17 3 hour 19 70701333 2 To view the name of the default system of units use the Units UsingSystem command 134 4 Basic Computations gt with Units gt UsingSystem SI To change the system of units use the Units UseSystem command gt UseSystem FPS gt 4 17 3 m 1 1 kg 666720741 10 f7 1b Extensibility You can extend the set of e Base dimensions and units e Complex dimensions e Complex units e Systems of units For more information refer to the Units AddBaseUnit Units AddDimension 2 Units AddUnit and Units AddSystem help pages For more information about units refer to the Units help page Scientific Constants and Element Properties Computations often require not only units see Units page 128 but also the values of sci entific constants including properties of elements and their isotopes Maple supports com putations with scientific constants You can use the built in constants and add custom con Stants Overview of Scientific Constants and Element Properties The ScientificConstants package provides the values of constant physical quantities for example the velocity of light and the at
302. s the type check Otherwise they return false Testing the Type of an Expression To test whether an expression is of a specified type 8 3 Working with Maple Expressions 349 e Use the type command gt type sin x trig rue gt type sin x cos x trig false For information on enclosing keywords 1n right single quotes see Delaying Evalu ation page 366 Maple types are not mutually exclusive An expression can be of more than one type gt type 3 constant rue gt type 3 integer rue For information on converting an expression to a different type see Converting page 356 Testing the Type of Subexpressions To test whether an expression has a subexpression of a specified type e Use the hastype command gt hastype sin x cos x trig true Testing for a Subexpression To test whether an expression contains an instance of a specified subexpression e Use the has command 350 8 Maple Expressions gt has sin x y x rie gt has sin x y x y true gt has sin x y sin x false The has command searches the structure of the expression for an exactly matching subex pression For example the following calling sequence returns false gt has x y z2z x 2 false To return all subexpressions of a particular type use the indets command For more inform ation see Indeterminates page 352 Accessing Expression Components Left and Right H
303. s you to move between placeholders Toolbar icons are controlled by the location of the cursor in the document For example place the cursor at an input region and the Text and Math icons are accessible while the others are dimmed See Table 1 4 for a list of the tools available in each icon Table 1 4 Toolbar Icons and their Tools Toolbar Icon Options Text tools Math Drawing Plot Animation C Text 7 Times Mew Roman 7 12 7 EZzU Math tools Text Math Drawing Plot C 2D Input Animation Yay BIU j 7 Times Hew Roman Drawing tools Text Math Plot Animation 1 1 Introduction to Maple 11 227x0009 MERDE 2 D Plot tools Text Math By ts Drawing Plot Animation HW RJA Oy Gi HH 3 D Plot tools Texl Math o 4 Drawing 45 4 t Animation tools Text Math 4 E gt pl Current Frame Drawing Table 1 5 Toolbar Icon Availability o ORA Os Ge Plot region Drawing and Plot icons Drawing Plot and Animation icons Canvas and Image regions Drawing icon The Text and Math icons allow you to enter text and math in the same line by choosing the appropriate input style at each stage when entering the sentence The derivative of sin x is cos x For an example see Example 6 Enter Text and 2 D Math in the Same Line page 31 Using the tools available in these icons you can customize the input style of the text and 2 D Math For the Text and Math
304. se the evalf command The evalf command returns a floating point or complex floating point number or expres alel gt evalf os ala 0 8660254040 gt eval a tod a 9 814954579 x x 23 14069264 gt evalf t 3 141592654 By default Maple calculates the result to ten digits of accuracy but you can specify any number of digits as an index that is in brackets gt evalf 40 1 3 141592653589793238462643383279502884197 For more information refer to the evalf help page 362 8 Maple Expressions See also Numerically Computing a Limit page 176 and Numeric Integration page 185 Evaluating Complex Expressions To evaluate a complex expression e Use the evale command If possible the evale command returns the output in the canonical form expr1 i expr2 In 2 D Math input you can enter the imaginary unit using the following two methods e In the Common Symbols palette click the i or j item See Palettes page 22 e Enter i or j and then press the symbol completion key See Symbol Names page 29 gt evalc J 1 Nie gt evalc sin 3 5j sin 3 cosh 5 Icos 3 sinh 5 In 1 D Math input enter the imaginary unit as an uppercase 1 I gt evalc 2 1 I 2 cos In 2 21 sin In 2 Evaluating Boolean Expressions To evaluate an expression involving relational operators gt lt lt and e Use the evalb command Note In 1 D Math input ente
305. see Evaluating Expressions page 67 From the context menu you can also select operations different from evaluation To the right of the expression Maple inserts a right arrow symbol and then the result For example use the Approximate operation to approximate a fraction 2 at 10 digits _ 3 0 606666666667 You can perform a sequence of operations by repeatedly using context menus For example to compute the derivative of cos x use the Differentiate operation on the expression and then to evaluate the result at a point use the Evaluate at a Point operation on the output and enter 10 evaluate at point differentiate w r t x JSI A 2sinf xs 20sin 100 cos x The following subsections provide detailed instructions on performing a few of the numerous operations available using context menus Figures in the subsections show related context menus or palettes Approximating the Value of an Expression To approximate a fraction numerically 1 Enter a fraction 2 Display the context menu See Figure 2 2 3 From the context menu select Approximate and then the number of significant digits to use 5 10 20 50 or 100 72 2 Document Mode Copy as MathML Paste Chrl Evaluate Evaluate and Display Inline Ctrl Explore Apply a Command Approximate pk o5 Assign to a Name 10 Denominator 20 b Numerator a0 100 Integer Functions b Units b 2 D Math b Figure 2 2 Appr
306. selected Enter Plot the ex pression Click the Math icon and enter the expression xX cos x Click the Text icon once again and enter and its derivatives 18 Click the text region and the border becomes highlighted You can now reshape the text region and move it around the plot region using the mouse xcos x cos x xsin x 2sin x xcos x Solution by Tutor The Student Calculus 1 package contains a tutor called Derivatives which displays a plot of the expression along with its derivatives In this example we solve the same problem as previously using this tutor 216 5 Mathematical Problem Solving 1 Load the Student Calculus 1 package Loading Student Calculus1 From the Tools menu select Load Package Student Calculus 1 2 Ctrl drag the expression xcos x to a blank document block region 3 Right click the expression and select Wren Toni Tutors Calculus Single Variable fie Hep Derivatives Note The Tutors menu is Plot window ee seo oe now available in the context menu because Fox hecost we loaded the Student Calculus 1 package a Fi b Fi in step 1 Derivatives F costse i x sin se In the Derivative Tutor the color swatch Serer E Display Fid in the plot shown beside the original expression 1s the PGA Fein costK color used for the curve in the plot region Display f x in the plot Similarly for f x and Jf x Sor
307. serere Ereren EENE E EEEE E EE 39 Tok Tp ea a REA A EAEE E E E EEE TEES 4 FAO fed IO NS 5 1S CAIN serenan nE ene anccnqesteacecsupennenesaceesaseepents eeaseeunabeaeiemsce 43 Me OTANI oso es soccer esos peeps ape EA E E 45 The Maple POLAT re aetcaecte ene naneaions vharendeaeiv lt eutaesiareeresieosanee lt uees uc eaeeseneetcens 45 Entering Commands Soar percep angus conta peneatemeaetiece demuestra nmneenciemircnnseenencnnieens 46 Document DIOCKS eee eee eee are ee Teer he eet ne Teter treet ee rte err ee rer 51 15 Ib Maple Help Sy SUC Ot seircarecieredsrinnat rinis eri Eke CEE SEn EENEN ECERS EEEREN 54 Accessing the Help SyYSIEME siccecssanegnrcenesasiiaesecdencaneaaaeacateneussuuienesnnnessens 54 Using the Help INARA OL ssrccertrceiris trinnin O aN A EAEAN 56 Viewing Help Paces as DOCUMENIS sriserosrrersessreresrsrostosrErn arar ESNA C ENAERE EE Erini 57 Viewing Examples in 2 D Math 2scccssinscssveesnsuvisssnnaticvesauscaseeeteneeeaseessueteresesss s COP n D E a E EE EEEE E EE E EA I L6 Available TCS OWE CS rereeeprren nre R A EE EA 58 Resources Available through the Maple Help System cccce cece ee ee ees 58 Maple Tour and Quick Resources ccc cece eee e cece ec eet eee ee eee eee ee ea een eee ee enees 59 NV CO OIL E E 6S OT E EIEE E T TA E E S 60 PADO a A OU E E E E ed E A E E E 63 PAA LO o EET EEE O E AA E E E EE E E 63 PAAL OLEE o EAN N EEE A A A E TE E TEA ET 63 2 3 NEMA Ci FEXpreSSiONS gpdnccsd artes ve nacinanten
308. serted command to display the plot by using the context menu item Evaluate gt plots interactive By default Maple displays each plot in a plot region using a different color You can also apply a line style such as solid dashed or dotted for each expression in the graph For more information refer to the plot options help page To see the Maple syntax used to generate this plot see Maple commands from Creating Plots Interactive Plot Builder page 250 Example 3 Display a plot of a multi variate expression Maple can display three dimensional plots and offers numerous plot options such as light models surface styles and shadings to allow you to customize the plot 6 2 Creating Plots 243 Launch the Interactive Plot Builder and enter an expression 1 Add the expression 1 sin x y x42 y 2 In the Select Plot Type window 2 Notice the available plot types for an expression with 2 variables as well as the plot objects for each type 3 Click Options In the Plot Options window 4 From the Variables column at the top of the dialog change the Range from field to 0 0 05 5 From the Label column enter z 6 From the Style group box select surface 7 From the Color group box in the Light Model drop down menu select green red 8 From the Color group box in the Shading drop down menu select z grayscale 9 From the Miscellaneous group box in the Grid Size drop down menu select 40 40 Plo
309. show all your calculations assumptions and any margin of error in your results You can also hide the computations to allow your reader to focus on the problem setup and final results The advanced formatting features lets you create the customized document you need Because the documents are live you can edit the parameters and with the click of a button compute the new results The Standard interface has two modes Document mode and Worksheet mode An interactive version of this manual is available in the Standard Work sheet interface From the Help menu select Manuals Resources and more Manuals User Manual Classic A basic worksheet environment for older computers with limited memory The Classic interface does not offer all of the graphical user interface features that are available in the Standard interface The Classic interface has only one mode Worksheet mode Command line version A command line interface for solving very large complex problems or batch processing with scripts No graphical user interface features are available Maplet Applications Graphical user interfaces containing windows textbox regions and other visual interfaces which gives you point and click access to the power of Maple You can perform calculations and plot functions without using the worksheet Maplesoft Graphing A graphical calculator interface to the Maple computational engine Using Calculator it you can perform simple c
310. siacsasweaverenssnesrmacostenunenenees 421 Sharing and Storing Maple Worksheet Content sosssensssesesrssersssrsseesss 423 x Contents List of Tables Table 1 1 Common Keystrokes for Entering Symbols and Formats 006 I Table 1 2 Maple Toolbar Options apices ccdncincswsareciawkckanenondanuns ce teawindeGecesnmnsninn ee heameweqiegne 9 Table L3 Tab Icon ID CSemp Ol sarrerei neinn einari e era EEEE EEE 10 Table 1 4 Toolbar Icons and their Tools scsccevssecscccavsevasserderdsprderseacevedserwecseegesveses 10 Table 1 5 Toolbar lecon Availability cosets cenaeaaceemtaceorrosacetenetetend omnes att qendewencasess 11 Table 1 6 Math Mode vs Text Mode oo nnnoonnnnenonnnnssennrssssreessssrrrssssreresssreessse 21 Toet PaE Ca OOS or A EEE oepaenee lt es 23 Table 1 39 Managing Palettes dese cvesse cute corte secetsapcamtcanenitastscatentector iecsboetens ests 25 Table 19 Help Page C70 ee ee ee eens eee 56 ToCA VOD COMMAS EEEE E EE E E E EE 84 TONS 2 TP Pa OC a E E EEEE AEE E E EAE E A 86 Table 4 1 Select Integer Commands cncccanaaeceecantsgaasnceqnsseesttantsecaaigevecsnaetenareetes 107 Table 4 2 Modular Arithmetic Operators ccccccecencec eee eneea ees eeeeeeaeeseeeeneeas 110 Table 4 3 Overview of Solution Methods for Important Equation Types 111 Table 44 amie DIMENSIONS serueriirisrrsi irra si Ern EE AR TET 129 Table 45 Scientine ONS AIS eerun eects ren EE EENE EEEE 135
311. sion gt assuming lt property or relation gt Properties and relations are introduced in The assume Command page 144 The frac command returns the fractional part of an expression gt frac x assuming x integer Using the assuming command is equivalent to imposing assumptions with the assume command evaluating the expression and then removing the assumptions gt about x As nothing known about this object If you do not specify the names to which to apply a property it 1s applied to all names gt gt assuming positive Sja a b Assumptions placed on names using the assume command are ignored by the assuming command unless you include the additionally option gt assume x lt 1 4 6 Restricting the Domain 147 gt is 1 gt 0 assuming x gt false gt is a gt 0 assuming additionally x gt 1 true The assuming command does not affect variables inside procedures For information on procedures see Procedures page 383 You must use the assume command gt f proc x sqrt a 2 x end proc f proc x sqrt a 2 x end proc gt f 1 assuminga gt 0 iF gt assume a gt 0 f1 a 1 For more information on the assuming command refer to the assuming help page 148 4 Basic Computations 5 Mathematical Problem Solving This chapter focuses on solving problems in specific mathematical disciplines The areas described below are not all that Maple provides
312. solve the problem For example when using context menus on Maple input in Document mode invoked by right clicking or Control clicking for Macintosh input and output are connected using an arrow or equal sign with self document ing text indicating the calculation that had taken place The command used to solve this expression 1s hidden Fre n n Anen Sgap When starting Standard Maple the default mode is Document mode Worksheet Mode Worksheet mode uses a Maple prompt as the default input region The Maple input prompt is a red angle bracket gt When using content menus on input in Worksheet mode all commands are displayed E eee yD gt solve x 2 4 7 x 10 0 x 2 h x 5 To work in Worksheet mode select File New Worksheet Mode 1 1 Introduction to Maple 5 Document and Worksheet Modes Regardless of which mode you are working in you have the opportunity to show or hide your calculations You can hide commands in Worksheet Mode by adding a document block from the Format menu Format Create Document Block see Document Blocks page 51 or you can show commands in Document mode by adding a Maple prompt from the Insert menu Insert Execution Group Before After Cursor see Input Prompt page 80 This chapter discusses features common to both modes Specific aspects of Document mode are explained in Document Mode page 63 and aspects of Worksheet mode are explained in Wo
313. sosssisseserissosesinsosisinstsrsirsv 332 Sclocing SS OUD occ aou sce necascn stan nae Er eN EEEE IEEE ENON EEPE NECETE NEEE RI 333 MS WACOM eeraa E E EENE E AE EE 333 1 9 Creating Graded ASSigimMeEnNiS 524 bance i peseusaanciestoraeceaansdecaessmusninpemiene eres 334 Creating SSO Ut srrrenesmerenek arieni Geneve sree sess peters E E stents etre alemdar ar 334 Viewing Questions m Maple sesrssrerirereercisisererkrro rebns Enn EEEE DEEE RES 334 Saving lest CONTENT sate rpendasnasctccceesnecosceneogewsuensocecenssemasostenisiriecesethese senses 335 1 10 Worksheet Compatibility ssc ccausaceansas seston reins acaecorne nn a 335 S Maple BP 4 6 Roki 6 5 a a en ner ERORE EAEE EEES 337 P E a Gee een ene E E E E ne 337 6 2 Creating and Using Data Structures sescicssrorirorriitesr radit ENS ois Ened iN Eska iNe 337 Expression SCQUCICES cscs sccposaincosndentsiicuianicndneinncoasieceayaielempaeslinvdimenceebioeennaa ea 338 DCIS E E E EE E E E eke E O EEE EEE EEEE 338 E e EE IOE A O ATA T A OO E OEE AEO EO EE EI A 339 ADS a A A E E A E E E 340 TaD e E E EE E E A E 342 e E e e a E EE E A A T E E A S 343 F ctonal OPEr lOrS siererirerrrir irr eee PE EENEN EET EEN eeeeeaees een teeceeee 343 SAD E E EE eg E eg E E EEEE ES 347 vill Contents 8 3 Working with Maple Expressions o ncrcds lt nc edvrnessdasecectvdanestaeeseeindinddcoseesecsed 348 Low Level Operations srssesiirisiscenieesissi eaiaieaedoeunew n EAEE EEE EEEE 348 Manipulating EXpPressio
314. srri Ciner ni Cisi rE EEO EEEN EEEN a eire nikia 307 Modifying the Structural Layout of a Table 20 0 0 ccc ccc ec ence neee een eneees 307 Modifying the Physical Dimensions of a Table ccc cece ccc ce eee ee eee en ees 310 Modifying the Appearance Of a Table occsccccsansassenciucioratecsnentansaiauinacenesecoass 310 PEA O E a EE E E EAE A AE 315 Execution Order Dapeng ene serrsrireniiriritiirisdi stirita dne Ka ETENEE EESE AR ones 315 Taplos and the Classic WORS DEE age cen cenonensassecaucanecisenevereasascusaceres esac owirras 315 Additional a 1616 ath in ee ee a nitre eee TENERSE ne eee eee 316 Po ANA AS ee asics se passe er can anteater acest EE N 318 inea Ge 10h eee ne eee nee E ee ee eee ee eT ee eee Troe rer ee 319 PGP aoc cas doc qon nes ese oes aoa A EEE E 320 AI Ee E OE EEA E T EEIE AEE 321 aAa e e e E EE E E E E E A TTET 322 Aeaee e E E A er E E ee A ETE ee 323 Inserting a Hyperlink in a Document seers ccsacte cee pen cercgcaveucnsveedogseie dee saeioanye ns 324 BOORDE aon een ne eee eee eer rene ee eet ere eee ar 327 Ta Embedded COMPONENTS scat piscceeecacciu sd acssstaoneteeseveminsy EEEE EEEE EEREN 329 Adding Graphical Interface Components ac0sc2csos3cescnensossanerescapnsssaanewemetsase 329 Task Template with Embedded Components ccccecesceceeeeneeeeeeeeeens 331 FS SPEIL C Ot across a scant a snsg ewe conan E rentns EE E E acne ETE 332 How to Use the Spellcheck Utility sss sririrssssrsrr
315. substitute le My Linear dlgebra BackwardSubstitute upper row echelon with options and overwrite Onear Algebra Backwardan Linear 4lgebra BackwardSubstitute upper row echelon with options Lneardigebra Backward suasd tute ME v ineardloehralRandMatriy Fram scalars Minas dAlcshral Randitatdr ii ted r 1 i gt 4 Some inserted commands have placeholders denoted by purple text The first placeholder is highlighted after you insert it into the document Replace it with your parameter then move to the next placeholder by pressing the Tab key Equation Labels Equation labels help to save time entering expressions by referencing Maple output See Figure 1 11 By default equation labels are displayed If equation labels are not displayed 1 From the Tools menu select Options and click the Display tab Ensure that the Show equation labels check box is selected 2 From the Format menu select Labels Ensure that both Execution Group and Worksheet are selected Figure 1 11 Equation Label 50 1 Getting Started To apply equation labels 1 Enter an expression and press Enter Note that the equation label is displayed to the right of the answer in the document 2 In a new execution group enter another expression that will reference the output of the previous execution group 3 From the Insert menu select Label Alternatively press Ctrl L Command L for Macintosh to open the Insert Label
316. t 2 D Math retains its format in both input and text regions When you copy and paste to another application in general Maple retains the original Structure 284 7 Creating Mathematical Documents Quick Character Formatting The Format Character menu provides access to the following quick formatting features Bold Italic Underline Superscript Subscript font Color and Highlight Color To modify text 1 In the document select the text to modify 2 From the Format menu select Character and then the appropriate feature For example in the pasted text select Calling Sequences and apply Bold character formatting Alternatively use the context bar icons For example to apply a color to the parameters f x xQ x1 Font Color Context Bar Icon 4 e Highlight Color Context Icon Br For font and highlight colors you can select from Swatches a color wheel RGB values or choose a color using the eye dropper tool See Figure 7 1 Flos Figure 7 1 Select Color Dialog In this example choose a dark purple color as in the help pages 7 2 Document Formatting 285 To format this text as bold click the Bold toolbar icon B Also select the text Calling Sequence and format as bold Result plot create a two dimensional plot Calling Sequence plotte x plottf x x0 x1 plotivl 72 Parameters f expression in independent variable x X mdependent variable X0 x1 left and ri
317. t 9 gt limit f x x infinity dim f x gt sum a k x k k 0 m product b j x j j 0 n Mn i gt a x b x LANG k 0 j 0 In Document Mode to enter input using Maple Input mode insert a Maple prompt by clicking in the toolbar and then click the Text button in the toolbar In Worksheet Mode simply click the Text button See Figure 1 2 Text Math Drawing Plot Animation C 2D Math Y Times Hew Roman w gl Figure 1 2 Text and Math buttons on the Toolbar 1 2 Entering Expressions 21 Table 1 6 Math Mode vs Text Mode Math Mode Text Mode Maple s default setting Executable standard Executable Maple notation This 1s also re math notation This is also referred to as 2 D ferred to as 1 D Math Input or Maple Input Math Input gt int x 2 2 x 1 gt xP 420414 i 3 L 3 2 i He Ps 3 When entering Maple Input or text in a text region highlighted in the toolbar Text or the Text mode icon is highlighted in the toolbar is entered in a document block with a slanted is entered with a vertical cursor as plain text cursor 2 f Ea Enter sorne text Palettes make entering expressions in familiar Using palettes while in 1 D Math teaches you the notation easier than entering foreign syntax and related Maple command syntax reduces the possibility of introducing typing er rors ow Expression z int t x If you prefer 1
318. t Li 3 symbol Li name lithium names lithium electronaffinity value 0 6180 uncertainty 0 0005 units eV atomicweight value 6 941 uncertainty 0 002 units amu boilingpoint value 1615 uncertainty undefined units K ionizationenergy value 5 3917 uncertainty undefined units eV density value 0 534 uncertainty undefined units a electronegativity value 0 98 uncertainty undefined units cm meltingpoint value 453 65 uncertainty undefined units K gt GetElement Li 4 Li massexcess value 25320 173 uncertainty 212 132 units keV bindingenergy value 4618 058 uncertainty 212 132 units keV atomicmass val ne 4 027182329 10 uncertainty 227 733 units namu 4 Value Units and Uncertainty To use constants or element properties you must first construct a ScientificConstants object To construct a scientific constant use the Constant command gt G Constant G To construct an element or isotope property use the Element command gt LiAtomicWeight Element Li atomicweight LiAtomicWeight Element Li atomicweight Value To obtain the value of a ScientificConstants object use the evalf command gt evalf G 1068912061 10 gt evalf LiAtomicWeight 2541006042 10 7 Note The value returned depends on the current system of units 138 4 Basic Computation
319. t Plot Type drop down menu select Animation 3 Change the x Axis range to 6 6 4 Change the y Axis range to 6 6 5 Change the Animation Parameter i range to 1 30 6 5 Creating Animations 271 In the Plot Options window 6 From the Style group box select surface 7 From the Color group box in the Light Model drop down menu select red turquoise 8 From the Color group box in the Shading drop down menu select z grayscale 9 In the View group box select the Constrained Scaling check box Plot the expression 10 Click Plot gt plots interactive For information on playing the animation see Playing Animations page 275 To see the Maple syntax used to generate this plot see Maple Syntax for Creating Animations Inter active Plot Builder Example page 271 The plots animate Command You can also use the animate command in the plots package to generate animations Table 6 5 The animate Command animate plotcommand plotarguments t a b animate plotcommand plotarguments t L e plotcommand Maple procedure that generates a 2 D or 3 D plot e plotarguments arguments to the plot command e t a b name and range of the animation parameter e t L name and list of real or complex constants To access the command use the short form name after invoking the with plots command gt with plots Maple Syntax for Creating Animations Interactive Plot Builder Example
320. t bar of your document 1 From the Format menu select Styles The Style Management dialog opens See Figure 7 4 To create a paragraph style e Click Create Paragraph Style The Paragraph Style dialog opens See Figure 7 6 e In the first row of the dialog enter a style name in the blank text field To modify a paragraph style e Select a paragraph style to modify Recall that all paragraph styles are preceded by the letter P e Click Modify The Paragraph Style dialog opens with the current attributes displayed For either action continue 4 In the Units drop down menu select the units used to determine spacing and indentation Select from inches in centimeters cm or points pt 294 e 7 Creating Mathematical Documents 5 Select the properties to use for this paragraph style such as Spacing Indent Alignment Bullets and Numbering Page Break Before and Linebreak 6 To add or modify a font style click Font The Character Style dialog opens For detailed instructions see Creating and Modifying Character Styles page 290 7 To save the style click OK or to abandon click Cancel If you are modifying an existing style all text in your document that uses the altered style is updated to reflect the changes Paragraph Style Properties Units pt Spacing Indent Line 0 0 Left Margin 0 0 Above 0 0 Right Margin 0 0 Below 0 0 First Line 0 0 Alignment Left Ise Bullets and Numbering Styl
321. t of the expression or the command used to generate the plot in the document To launch the Interactive Plot Builder e From the Tools menu select Assistants and then Plot Builder Note The Tools menu also offers tutors to easily generate plots in several academic subjects For more inform ation see Teaching and Learning with Maple page 198 Table 6 1 Windows of the Interactive Plot Builder 1 Specify Expressions window 2 Select Plot Type window Interactive Plot Builder Specify Expressions Ea Mi interactive Plot Builder Select Plot Type File Select Plot Type and Functions Expressions Plot iy Edit Functions sini y ile e y 1 Select Plot Edit 3 D plot 3 D contour plot 2 D contour plot 2 D gradient vector field plot 2 D density plot 2 D implicit plot Select Variable Purposes Ranges and Plot Options Variables y y Axis S On Plot return plot command a toe Axis ts 240 6 Plots and Animations 1 Specify Expressions window Add edit or remove expressions and variables Once finished you can advance to the Select Plot Type window 2 Select Plot Type window Select the plot type and corresponding plot and edit the ranges Once finished you can display the plot or advance to the Plot Options window 3 Plot Options window x 3 D Plot plot 3d Variables Label Orientation x horizontal 3 horizontal l Range from horizontal ha l
322. t of typing when entering Maple commands you can use command completion Command completion displays a list of all Maple packages commands and functions that match the entered text If there are multiple ways to call a command then the command completion list contains each one with appropriate placehold ers To use command completion 1 Begin entering a command or package name 2 Select Tools Complete Command or use the shortcut key Ese see Shortcut Keys by Platform page xiv If there is a unique completion it is inserted Otherwise a list of possible matches is displayed 1 4 Commands 49 3 Select the correct completion from the list finer Linear Algebra Linear lgebra pa Linear algebra dd linear combination UnearAleebra Add Mul Mv a Linear algebral dd linear combination with scalars and constructor options Lineardigebra Add liv v2 xl x2 Linear Algebral dd linear combination with scalars GnearAleebra Add Ji dvi Mve xl x2 Linear Algebral4dd linear combination with scalars constructor options and overwrite LinearAlgebra Add i v Linear Algebral4Adjaint square Matrix OnearAleebra Adtotnt A Linear AlgebralAdjaint square Matrix with constructor options Lineardigebra Adjant Jd oufpufopdons ist Linear Algebra BackwardSubstitute tupper row echelon GnearAlcebra Fachwardswbsd futa Aa Linear Algebra BackwardSubstitute upper row echelon GnearAleebra Fackward
323. t the expression 10 Click Plot To see the Maple syntax used to generate this plot see Maple commands from Creating Plots Interactive Plot Builder page 250 Example 4 Display a conformal plot Maple can display a conformal plot of a complex expression mapped onto a two dimensional grid or plotted on the Riemann sphere in 3 D Launch the Interactive Plot Builder and enter an expression 1 Add the expression z 3 In the Select Plot Type window 2 From the Select Plot group box select 2 D conformal plot of a complex valued function 3 Change the range of the z parameter to 0 2 2 I 244 e 6 Plots and Animations In the Plot Options window 4 From the Axes group box select normal 5 From the Miscellaneous group box select the Grid Size drop down menu option 30 30 Plot the expression 6 Click Plot Example 5 Display a plot in polar coordinates Cartesian ordinary coordinates is the Maple default Maple also supports numerous other coordinate systems including hyperbolic inverse elliptic logarithmic parabolic polar and rose in two dimensions and bipolar cylindrical bispherical cylindrical inverse elliptical cylindrical logarithmic cosh cylindrical Maxwell cylindrical tangent sphere and toroidal in three dimensional plots For a complete list of supported coordinate systems refer to the coords help page Launch the Interactive Plot Builder and enter an expression 1 Add the expression
324. t the expression 2 Press Ctrl Command for Macintosh or Enter The result is updated To update a group of computations 1 Edit the expressions 2 Select all edited expressions and the results to recalculate 2 6 Performing Computations 69 3 Click the Execute toolbar icon All selected results are updated To update all output in a Maple document e Click the Execute All toolbar icon M All results in the document are updated 2 6 Performing Computations Using the Document mode you can access the power of the advanced Maple mathematical engine without learning Maple syntax In addition to solving problems you can also easily plot expressions The primary tools for syntax free computation are e Palettes e Context menus e Assistants and tutors Note The Document mode is designed for quick calculations but it also supports Maple commands For information on commands see Commands page amp 2 in Chapter 3 Worksheet Mode page 79 Important In Document mode you can execute a statement only if you enter it in Math mode To use a Maple command you must enter it in Math mode Computing with Palettes As discussed in Entering Expressions page 64 some palettes contain mathematical oper ations To perform a computation using a palette mathematical operation 1 In a palette that contains operators such as the Expression palette click an operator item 2 In the inserted item specif
325. tarted Don t worry about your difficulties in Mathematics I can assure you mine are still greater Albert Einstein Mathematics touches us every day from the simple chore of calculating the total cost of our purchases to the complex calculations used to construct the bridges we travel To harness the power of mathematics Maplesoft provides a tool in an accessible and com plete form That tool is Maple In this chapter Introduction to Maple page 2 The main Starting the Standard Document Interface features of Maple s Standard Interface Entering commands and mathematical expres sions Toolbars Context menus Copy and drag keys Saving Maple documents Entering Expressions page 19 Methods of Execution groups entering expressions in 1 D and 2 D Math Watiiode and Text Mode Palettes Symbol names Toolbar icons Point and Click Interaction page 32 Anintro Assistants duction to the point and click features in Maple Tutors Context menus Task templates Exploration Assistant Commands page 45 An introduction to the Using commands from the Maple library commands of the Maple language Entering commands Document blocks 2 Getting Started The Maple Help System page 54 Accessing How to access help for Maple features help on commands packages point and click Interacting with help pages features and more Viewing and interacting with examples Available Resources page 58 Both onl
326. tatement sequence else statement sequencell end if The conditional expressions conditional_expression1 conditional_expression2 can be any boolean expression You can construct boolean expressions using e Relational operators lt lt gt gt lt gt e Logical operators and or xor implies not e Logical names true false FAIL The statement sequences statement_sequence1 statement_sequence2 statement_sequen ceN can be any sequence of Maple statements including if statements The elif clauses are optional You can specify any number of elif clauses The else clause is optional 9 2 Flow Control 371 Simple if Statements The simplest if statement has only one conditional expression if conditional expression then gLat emant sequence end if If the conditional expression evaluates to true the sequence of statements is executed Otherwise Maple immediately exits the if statement For example gt x 1173 gt if not isprime x then ifactor x end if EIJ 2 Led t3 ld else Clause In a simple if statement with an else clause if the evaluation of the conditional expressions returns false or FAIL Maple executes the statement sequence in the else clause For example gt if false then if statement else else statement end if else statement elif Clauses In an if statement with elif clauses Maple evaluates
327. te see Figure 5 2 When creating a Matrix using the Matrix command there are several input formats available For example enter a list of lists The dimensions of the matrix are inferred from the number of entries given gt Matris 1 m 0 e sin t m 0 0 sell TU Q 7 87 v i Se tti ar Alternatively use the angle bracket shortcut lt gt Separate items in a column with commas and separate columns with vertical bars gt 1 Tc O e sin f Pat 0 se T sin t O of 0 a Jg For information on the Matrix command options see Creating Matrices and Vectors with Specific Properties page 164 160 5 Mathematical Problem Solving Use the Matrix palette to interactively create a matrix without commands OW Matrix Rows Columns Choose Type Custom values 7 Shape Any Data type Any FE Insert Matrix Figure 5 2 Matrix Palette In the Matrix palette you can specify the matrix size see Figure 5 3 and properties To insert a matrix click the Insert Matrix button 5 3 Linear Algebra Rows Columns 3x S CSKIFE x10 Control x100 Figure 5 3 Matrix Palette Choosing the Size After inserting the matrix 1 Enter the values of the entries To move to the next entry placeholder press Tab 2 After specifying all entries press Enter L e am m sin t O 0 5e Creating Vectors You can create a Vec
328. te command 271 contourplot command 260 display command 262 matrixplot command 258 pointplot command 257 series 183 Statistics 195 viewing animations animate context bar 275 point and click 32 polynomial equations 436 Index solving 116 numerically 117 polynomials algebra 150 arithmetic 150 coefficients 156 collecting terms 156 degree 156 division 150 152 efficient arithmetic 158 expanding 151 factoring 157 implied multiplication 152 numeric algebraic manipulation 158 operations 157 sorting 152 pure lexicographic 153 total degree 153 PolynomialTools package 158 IsSelfReciprocal command 158 powers entering 6 precalculus teaching 199 precision 104 prem command 158 previously assigned 366 primality testing 108 primpart command 158 print command 385 table 315 printing embedded components 392 probability distribution 193 proc key word 383 procedures 385 and assumptions 147 calling 383 defining 383 displaying 385 inputs 384 multiple lines 384 output 385 using 383 product command 381 products entering 6 implied 6 programming access to Maple s programming guides 59 programs 369 modules 386 procedures 385 prompt input 80 properties testing 145 protected names 95 Q QPSolve command 191 QR factorization 173 quadratic programs 191 quantities with uncertainty 140 accessing error 140 accessing value 140 computing with 141 constructi
329. tent Insert Minimal Content Insert into New Worksheet _ Display task markers Volume of Revolution Description Calculate the volume of revolution for a solid of revolution when a function is rotated about the horizontal or vertical axis Enter the function as an expression and specify the range gt sin x cos x 1 0 sin x cos x 1 0 Calculate the volume of revolution gt Student Calculus 1 VolumeOfRevolution 1 g 7 vga eer iL i6 Display the floating point value using the evalf command bi evalf 2 8 693245131 Display a plot using the output pilot option gt Sfudeni Calculus I VolumeOfRevolition 1 output plot scaling constrained title For details on inserting and using task templates see Task Templates page 41 You can also create your own task templates for performing common tasks For details refer to the creatingtasks help page 3 8 Text Regions 93 3 8 Text Regions To add descriptive text in Worksheet mode use a fext region To insert a text region e In the toolbar click the Text region icon T The default mode in a text region is Text mode In a text region you can e Enter text with inline mathematical content by switching between Text and Math modes To toggle between Text mode and Math mode press F5 or click the Math and Text toolbar icons Math Note The mathematical content in a text region is not evaluated To
330. ter Components and Maplets page 389 332 e 7 Creating Mathematical Documents 7 8 Spell Checking The Spellcheck utility examines all designated text regions of your document for potential spelling mistakes including regions that are in collapsed sections It does not check input output text in execution groups or math in text regions See Figure 7 23 Note The Spellcheck utility uses American spelling CodeGeneration package is a collection of comands and subpackages that enable the translation of Maple code to other programming languages Spellcheck Mot Four comanda Change To commands Suggestions Ignore All Change All Figure 7 23 Spellcheck Dialog How to Use the Spellcheck Utility 1 From the Tools menu select Spellcheck Alternatively press F7 The Spellcheck dialog appears It automatically begins checking the document for potential spelling mistakes 2 If the Spellcheck utility finds a word that it does not recognize that word is displayed in the Not Found text box You have six choices e To ignore the word click Ignore e To ignore all instances of the word click Ignore All e To change the word that is accept the suggested spelling that is in the Change To text box click Change e To change all instances of the word that is accept the suggested spelling to replace all instances of the word click Change All 7 8 Spell Checking 333 e To add the word to your dictionary
331. th Examples 219 2 Right click this equation and select Manipulate B isiteiinipaaier Equation The Manipulate Equation dialog displays Group all of the terms to the left 3 In the Addition region the Group terms row allows you to group terms on a specified side With the left side already selected click Do Expand the left side of the equation 4 In the Miscellaneous Operations region we can manipulate the equation by applying a com mand from the drop down menus Since we want to expand the left side of the equation only click the first drop down menu in the second row and select expand Click Do Factor the equation 5 From the same drop down menu select factor and click Do x Th ee 24 ee 1 A 4 I Bh sepa stacked varacak Pis Saure both sides Te ppa root of both sacle Plates bih sacle to power 3 Exponentiete bath sides using base Z Pispalan Cpe ation Apply en os to both sides Apply amy ow to eee e ne Sa Compite the sume on the left sde a i Cancel Equation Manipulator x 7 x 1 4 x 1 4 x 4 x 7 3 x 1 4 x 4 0 C Show steps stacked vertically E1 x 7 2 x 1 2 4 x 1 2 4 x 4 2 E2 lhs E1i rhs Ei 0 Power Square both sides Take square root of both sides Raise both sides to power 3 Group terms on left Miscellaneous Operations Apply Ra to both sides pool expand v to eft side exp 220 5 Mathematica
332. the substitution This is the most common use of the eval command For example substitute x 3 in the following polynomial gt x 4x 7x4 2 49 Fx4 2 gt eval x 4x 7x 2 x 3 44 To substitute a value for a variable using palettes HEJ 1 In the Expression palette click the evaluation at a point item o 360 8 Maple Expressions 2 Specify the expression variable and value to be substituted For example gt ae x 5 J 17 Substitutions performed by the eval function are syntactical not the more powerful algeb raic form of substitution If the left hand side of the substitution is a name Maple performs the substitution gt eval cos abc a 4 cos 4 th J 6 If the left hand side of the substitution is not a name Maple performs the substitution only if the left hand side of the substitution 1s an operand of the expression gt eval cos ab ab gt eval cos abc ab JE cos a bc Maple did not perform the evaluation because ab is not an operand of cos abc For in formation on operands refer to the op help page For algebraic substitution use the algsubs command or the simplify command with side relations 8 3 Working with Maple Expressions 361 gt algsubs ab S cos a be l os en cos 4 gt simpli cos abc ab ala rt gt 0s TER cos 4 Numerical Approximation To compute an approximate numerical value of an expression e U
333. the Unit Converter or the convert units com mand gt convert 450 2 kg units Ib 992 5211043 Zb For information on the Unit Converter and using units see Units page 128 Convert a list to a set 8 3 Working with Maple Expressions 357 gt convert a b c d set l 23 c d 4 5 6 Array 1 2 2 8 9 2 3 3 5 2 4 2 4 2 5 Fortran order Lad b gt im iy pH Il Md ee Il t wo M as I on TE Il d E bd ke I uD H m J 4 C Sa m mai 1 as Il Sa i a md ez oq Da P a s S HE ma Ta Cr h a as Ti He m Es iy T a at T Ti Maple has extensive support for converting mathematical expressions to a new function or function class gt convert cos x exp t a j tu lan Find an expression equivalent to the inverse hyperbolic cotangent function in terms of Le gendre functions gt convert arccoth z Legendre LegendreQ 0 ri 3 z For more information on converting to a class of functions refer to the convert to_spe cial function help page Normalizing To normalize an expression e Use the normal command The normal command converts expressions into factored normal form gt normal f y x y Ca ETY x y You can also use the normal command for zero recognition 358 8 Maple Expressions gt normal x 1 x
334. the statement se quence in the else clause gt x 12 9 2 Flow Control 373 gt if not type x integer then printf ca is not an integer x elif x gt 10 then printf Sa is an integer with more than one digit x elif x gt 0 then printf Sa is an integer with one digit x else printf Sa is a negative integer x end if 12 is a negative integer For more information on the if statement refer to the if help page Repetition for Statement Using repetition statements you can repeatedly execute a statement sequence You can repeat the statements in three ways e Until a counter variable value exceeds a limit for from loop e For each operand of an expression for in loop e Until a boolean condition does not hold while loop for from Loop The for from loop statement repeats a statement sequence until a counter variable value exceeds a limit Syntax The for from loop has the following syntax for counter from initial by increment to final do statement sequence end do The behavior of the for from loop 1s 1 Assign the initial value to the name counter 2 Compare the value of counter to the value of final If the counter value exceeds the final value exit the loop This is the loop bound test 3 Execute the statement_sequence 374 9 Basic Programming 4 Increment the counter value by the value of increment 5 Repeat steps 2 to 4 until Mapl
335. tinuously while Dragging check box is selected 396 10 Embedded Components and Maplets The value from the slider as you move the arrow indicator populates the Label caption field For details on this command refer to the 7 DocumentTools Do help page Example 2 Creating Embedded Components In chapter 7 see Embedded Components page 329 you created a document that included embedded components imported from a task template Here we re create that configuration of components This example takes two parameters a and b as inputs then plots the function y bx a and calculates 7 1 Create the components The table layout is best done after the components are finished in case the configuration of the components changes as you are working Create two DialComponents to set the parameters a and b one GaugeComponent to display the result F7 one PlotComponent to display the plot and one MathContainer Component to display the function Note that you do not need to use the dial and gauge components here there are others such as the slider that could also be used 10 3 Creating Embedded Components 397 Figure 10 4 The Inserted Components 2 Edit the display of the components Open the Component Properties dialog for the first DialComponent and notice that it already has a name This name is used to reference the component from other components and is unique Change the display of each of the compo
336. tional operator x x 1 tof a Tae i all Assign the expression x 1 tog Fe ie tn ae To evaluate the functional operator f at a value of x e Specify the value as an argument to f 8 2 Creating and Using Data Structures 345 gt f 22 To evaluate the expression g at a value of x e You must use the eval command gt g 22 gt eval g x 22 For more information on the eval command and on using palettes and context menus to evaluate an expression at a point see Substituting a Value for a Subexpression page 359 Multivariate and Vector Functions To define a multivariate or vector function e Enclose coordinates or coordinate functions in parentheses For example a multivariate function p gt f x y gt ce gt f 0 0 7 2 1 1 9 008893709 A vector function gt g t sin t cos t t 346 8 Maple Expressions t d OR z Using Operators To perform an operation on a functional operator specify arguments to the operator For example for the operator f specify f x which Maple evaluates as an expression See the following examples Plotting Plot a three dimensional operator as an expression using the plot3d command gt h x y gt xcos y gt plot3d h x y x 2 2 y 2 1 27 For information on plotting see Plots and Animations page 237 8 2 Creating and Using Data Structures 347 Integration Integrate a function using the int
337. tions exponential functions powers and various special functions You can also specify custom simplification rules using a set of side relations i gt Pa 3 34 2 gt simplifv sin x In 2y cos x 1 In 2 In y To limit the simplification specify the type of simplification to be performed 354 8 Maple Expressions gt simplify sin x In 2y cos x trig 1 In 2 y gt simplifv sin x In 2y cos x In sin x In 2 In y cos x You can also use the simplify command with side relations See Substituting a Value for a Subexpression page 359 Factoring To factor a polynomial e Use the factor command gt factor x 93 x 2027 12x e x 2 3 9 t IP gt factor x y x 3x x y 2xy 6x 5xy y 3x 3y 3 Pee LY ey Maple can factor polynomials over the domain specified by the coefficients You can also factor polynomials over algebraic extensions For details refer to the factor help page For more information on polynomials see Polynomial Algebra page 150 To factor an integer e Use the ifactor command gt ifactor 196911 3 11 13 17 For more information on integers see Integer Operations page 106 Expanding To expand an expression 8 3 Working with Maple Expressions 355 e Use the expand command The expand command distributes products over sums and expands expressions within function
338. tnsvacerovessuinaiestacwsirscdcedesceension ties 262 6 CSO 1 PIOS prster iseer enr EE ER EE E EEEE 264 Interactive Plot Builder OPHONS secpesasunsat cis vecrewacnenceeeaune EEEREN ENN 264 Context Men TIONS eeces cig nen cesses haretene nore desea EE E EREE 264 The plot and plot3d MUONS sisisisicssrioniscsisnssisissisdoisdsidicessrdsosiiedidesirisdteni 267 6 4 Analyzing Plots serseresnreriderieri sensorien ENEE OaE EEE Ern EEr EEEa 269 Point Probe Rotate Pan and Zoom Tools 20 20 00 ccc cece cece ee se cee esececeeeeeeenees 269 Os LC ALG AMANOS garra ete neces ee cade EEA AT 270 Interactive Plot BUlder srrnirerersereinrrerertonncirr tne EErEE EE ARERR EEE EENEI RETEN EE 270 The plots animate Command noooonnesennnnessnnnsssseneessserersssreersssrresssene 271 The plotsdl view point Command 2 2240 sat erisecnenssasieoeraciadurcs eelesiehaniacdanies 273 06 Playme Aa iOis 2caeneesncacveccotasesavactscsaaetoncseeeeneasenecacumareatenresunee 275 Animation Context Bat cccievccice lt cinnscsassasenasentenexdesevbedusew nana dewantieenadeeesseees 21D 67 C stomizine ANIMANONS nacuccscarcscasaisnacnveracoiesasnsaverseqsnemepeecsdatecemtaseneens 276 Interactive Plot Builder Animation Options scciessesisvevsessestavicwensseseeseesestens 276 Context Menu Options gin cacexccceeentaceueaconsanicakepmasre essa cueunoceneeaneeneasocs 2T TG ai inate Command OOS deze vc pees de ernie csasince docx eae eae sen edeasaee 211 CS EDO e E A E E EE RE S 219 6
339. to Vera C Bold Agency FB ai Jal taic _ Underlined I Arial Black Superscript Arial Marrow C Subscript lA rial Rounded MT Bold Arial Unicode M5 Baskerville Old Face Bauhaus 93 Bell MT Berlin Sans FB Berlin Sans FB Demi Maplesoft Figure 7 5 Character Style Dialog For example in the pasted text suppose we want to create a character style for the bold purple parameter e From the Format menu select Styles then click Create Character Style e Enter the style name Placeholder and then select the character attributes In this case click the Bold check box Then click the Color button and choose a dark purple Click OK to create the character style Now you can apply the style to any text Under Calling Sequences select each list of parameters inside the command To apply the style from the Styles drop down menu in the toolbar select Parameter C Equation Label C Header and Footer C Hyperlink C Maple Input C Maple Input Placehe C Page Number C Parameter C Text 292 e 7 Creating Mathematical Documents Result plot create a two dimensional plot Calling Sequence plottf x plottf x x0 x1 plotivl v2 Parameters f egpression in independent variable x F independent variable 0 1 left and right endpoimts of horizontal range i v2 x coordinates and y coordinates Applying Paragraph Styles By using the drop down list in t
340. to and select one of the assistants or tutors For more information on assistants and tutors see Assistants page 32 in Chapter 1 3 7 Task Templates Maple can solve a diverse set of problems The task template facility helps you quickly find and use the commands required to perform common tasks After inserting a task template specify the parameters of your problem in the placeholders and then execute the commands or click a button The Task Browser Figure 3 4 organizes task templates by subject To launch the Task Browser e From the Tools menu select Tasks and then Browse You can also browse the task templates in the Table of Contents of the Maple Help System 92 e 3 Worksheet Mode File view E Algebra H Calculus Differential eo Calculus Integral o E Integration He 4oproximake Integration E E Methods of Integration o pplications f aes Arc Length of a Univariate Fu ee a Solids of Revolution H Surface Area T Volume BS Series H E Calculus Multivariate H Calculus Vector sue Convert Expression to Function E E Curve Fitting H E Differential Equations E E Document Templates H Evaluating H E Geometry H E Integers H Linear Algebra H E Lists E E Maple T A E E Plots H E Polynomials H E Statistics H E Transformations ae Units Constants and Errors F3 jiii ask VolumeOrReyUnivariateFcn Figure 3 4 Task Browser Copy Task to Clipboard Insert Default Con
341. to the plot device help page 6 9 Code for Color Plates Generating impressive graphics in Maple can require only a few lines of code as shown by the examples in this chapter However other graphics require many lines of code Code for the color plates 1s available at the Maple Application Center From the Help menu select On the Web User Resources and then Application Center To access the color plate code 1 Go to the Maple Application Center 2 In the Keyword or phrase region enter Color Plate 280 6 Plots and Animations 7 Creating Mathematical Documents Maple allows you to create powerful documents as business and education tools technical reports presentations assignments and handouts You can e Copy cut and paste information e Format text for reports or course material e Add headers and footers e Insert images tables and symbols e Generate two and three dimensional plots and animations e Sketch in the document or on a plot e Insert hyperlinks to other Maple files web sites or email addresses e Place instructions and equations side by side e Bookmark specific areas e Easily update revise and distribute your documents In this chapter we will create a document that demonstrates many of Maple s documentation features For further examples note that this guide was written using Maple 7 1 In This Chapter Document Formatting page 283 Add Copy and Paste page 283 various text fo
342. togram e Pie chart e Scatter plot S 27x For example create a scatter plot for a distribution of points that vary from sin 500 by a small value determined by a normally distributed sample gt N 200 V U Sample Normal 0 1 N gt X lt seq x x 1 N gt V HEL i w gt 2 Y lt seq sin 5 V ScatterPlot X Y title Scatter Plot Scatter Plot To fit a curve to the data points include the optional fit equation parameter Using the plots display command create a plot that contains 5 6 Statistics 197 e a scatter plot of the data points e a quartic polynomial fitted to the data points f x ax hbxe cex dx t e BEX e the function sin zz gt P ScatterPlot X Y fit ax b tex dx e x thickness 2 LK gt 0 plo sin x 1 N thickness 2 color red gt plots display P Q title Scatter Plot with Fitted Quartic Polynomial Scatter Plot with Fitted Quartic Polynomial gt For more information on statistical plots refer to the Statistics Visualization help page For an overview of plotting see Plots and Animations page 237 198 e 5 Mathematical Problem Solving Additional Information For more information on the Statistics package including regression analysis estimation data manipulation and data smoothing refer to the Statistics help page The Data Analysis Assistant provides an interactive wa
343. tor using angle brackets lt gt 161 162 e 5 Mathematical Problem Solving To create a column vector specify a comma delimited sequence lt a b c gt The number of elements is inferred from the number of expressions gt 1 2 3 bo Cae To create a row vector specify a vertical bar delimited sequence lt a b c gt The number of elements is inferred from the number of expressions gt 1 2 3 For information on the Vector command options refer to the Vector help page You can also create vectors using the Matrix palette If either the number of rows or number of columns specified is 1 then you have the option of inserting a matrix or inserting a vector of the appropriate type See Figure 5 4 Columns Choose Type Custom values Y Shape Any Data type Any Insert Matrix Insert Vector row Insert Vecto r column n o Figure 5 4 Insert Matrix or Insert Vector Viewing Large Matrices and Vectors Matrices 10 x 10 and smaller and vectors with 10 or fewer elements display in the docu ment Larger objects are displayed as a placeholder For example inserta 15 x 15 matrix 5 3 Linear Algebra e 163 In the Matrix palette 1 Specify the dimensions 15 rows and 15 columns 2 In the Type drop down list select a matrix type for example Random 3 Click Insert Matrix Maple inserts a placeholder 15 x 15 Matrix Data Type a
344. u aresin gt x 9 Right click the output and select Solve Solve fh for Variable u A 8 The solution is aresin al Example 6 Initial Value Problem Solve and plot the solution of the initial value problem y t 4y t 13 y t cos 2t y 0 2 y 0 1 234 e 5 Mathematical Problem Solving Solution by ODE Analyzer Assistant The ODE Analyzer Assistant lets you solve ODEs numerically or symbolically and displays a plot of the solution Action Result in Document 1 Enter the ODE in a blank document y t 4y t 13 y t cos 2 2 block region 2 Right click the equation and select WODE Analyzer Assistant Solve DE Interactively The ODE Ana biferential Equations Conditions Parameters lyzer Assistant displays with the ODE x 0 4y t 13y cas 2 automatically inserted To insert the initial conditions 3 In the Conditions region click Edit Add Condition The Edit Conditions dialog opens at Edit Conditions 4 In the Add Condition region with y Add Condition selected in the drop down menu enter 0 at in the first text field to the right and 2 in l the second text field Click Add Your entry should match the one shown to the 0 2 right Edit Conditions 5 8 Clickable Math Examples 235 5 To enter the initial condition for Differential Equations Conditions y select y from the drop down y t Ay li 13 y t cos 2 2 A
345. unctions written in these languages can be linked and used as if they were Maple procedures With external calling you can use pre written optimized algorithms without the need to translate them into Maple commands Access to the NAG library routines and other numerical al gorithms is built into Maple using the external calling mechanism External calling can also be applied to functions other than numerical algorithms Routines exist that accomplish a variety of non mathematical tasks You can use these routines in Maple to extend its functionality For example you can link to controlled hardware via a serial port or interface with another program The Database package uses external calling to allow you to query create and update databases in Maple For more information refer to the Database help page 11 5 Connectivity 421 For more information on using external calling refer to the ExternalCalling help page Mathematica Translator The MmatTranslator package provides translation tools for converting Mathematica ex pressions command operations and notebooks to Maple The package can translate Math ematica input to Maple input and Mathematica notebooks to Maple documents The Mma subpackage contains commands that provide translation for Mathematica commands when no equivalent Maple command exists In most cases the command achieves the translation through minor manipulations of the input and output of similar Maple commands
346. ur document Editing Components Removing Components Integrating into a Document Using Maplets page 400 Methods for launching a Maplet File Maplet Maple Document Authoring Maplets page 402 Methods for authoring Maplet Builder and saving a Maplet Maplets Package Saving 10 2 Using Embedded Components Interacting Embedded components allow readers to interact with Maple code through graphical com ponents rather than commands They can be used alone as with a button that you click to execute code or together such as a drop down menu where you select an item and a change takes place in a plot component Component Descriptions Table 10 1 Embedded Component Descriptions Component Name and Description Inserted Image Button Click to perform an action that is execute code 389 390 10 Embedded Components and Maplets Component Name and Description Inserted Image Check Box Select or de select Change the caption and Jj check ox enter code to execute when the value changes Combo Box Select one of the listed options from the drop down menu Change the items listed and enter code to execute when the value changes Dial Select or display an integer or floating point value Change the display and enter code to execute when the value changes ny Label Display a label The value can be updated based Label on code in the document or another embedded component List Box Display
347. ure 11 1 Import Data Assistant 11 3 Reading from Files 415 ImportMatrix Command The Import Data Assistant provides a graphical interface to the ImportMatrix command For more information including options not available in the assistant refer to the Import Matrix help page Reading Expressions from a File You can write Maple programs in a text file using a text editor and then import the file into Maple You can paste the commands from the text file into your document or you can use the read command When you read a file with the read command Maple treats each line in the file as a com mand Maple executes the commands and displays the results in your document but it does not by default insert the commands from the file in your document For example the file ks txt contains the following Maple commands S n gt sum binomial n beta 2 beta 2 beta beta beta beta l1 n S 19 Note that the file should not contain prompts gt at the start of lines When you read the file Maple displays the results but not the commands S n gt binomial n B en aP 1024937361666644598071 1 14328769317982974 11 6 gt filename cat kernelopts datadir kernelopts dirsep ks txt gt read filename Error unable to read C Program Files Maple 14 data ks txt If you set the interface echo option to 2 Maple inserts the commands from the file into your document 416
348. ve of xsin ax x with respect to x 2 gt x sin ax x 2 cos ax a xsin ax y 2 To calculate the mixed partial derivative of xsin 3v y X 178 5 Mathematical Problem Solving 2 5 ay ax x sin 3 vy yx 3cos 3y 5x Note To enter another 0 symbol you can copy and paste the existing symbol or enter d and use symbol completion The diff Command Maple computes derivatives using the diff command To directly use the diff command specify the expression to differentiate and the variable gt xsin ax x xsin ax x 5 1 gt diff 5 1 x sin ax xcos ax a 2x 5 2 For information on equation labels such as 5 1 see Equation Labels page 97 You can calculate a higher order derivative by specifying a sequence of differentiation variables Maple recursively calls the diff command gt diff C5 1 x x 2cos ax a xsin ax r 3 5 3 To calculate a partial derivative use the same syntax Maple assumes that the derivatives commute gt diff xsin 3y y x x y l 24x To enter higher order derivatives it is convenient to use the syntax diff f x n This syntax 3cos 3 y can also be used to compute the symbolic n order derivative 5 4 Calculus 179 For example gt diff cos t t n cos nt i Differentiating an Operator You can also specify a mathematical function as a functional operator a mapping For a comparison
349. w or equal sign indicating the action that had taken place For more information see Context Menus page 39 Copy amp Drag With Maple you can drag input output or curves in a plot region into a new input region This is done by highlighting the input or selecting the curve and dragging it with your mouse into a new input region Dragging the highlighted region will cut or delete the original input To prevent this use the copy and drag feature e Ctrl drag Windows and UNIX e Command drag Macintosh That is highlight the region you want to copy Press and hold the Ctrl key while you drag the input to the new region using the mouse The steps are the same for Macintosh with the exception of pressing the Command key 1 1 Introduction to Maple 13 Example 2 Solve and Plot an Equation Using Context Menus and Copy amp Drag Review the following example 5 7 3x t2 In this example we will enter the equation and then solve and plot the equation using context menus and Maple s copy amp drag feature This example will only refer to the keystrokes needed on a Windows operating system to invoke the context menus and the copy amp drag feature For your operating system refer to section Shortcut Keys by Platform page xiv for the equivalent keystrokes 14 1 Getting Started To solve the equation 1 Enter the equation 2 Right click the equation and select Move to Left Input gt Se 7 3x4 2 Copy as
350. where x is a vector of problem variables possibly subject to constraints input is in equation or Matrix form Maximize Compute a local maximum of an objective function possibly subject to constraints Minimize Compute a local minimum of an objective function possibly subject to constraints NLPSolve Solve a non linear program NLP which involves computing the minim um or maximum of a real valued objective function possibly subject to constraints input is in equation or Matrix form QPSolve Solve a quadratic program QP which involves computing the minimum or maximum or a quadratic objective function possibly subject to linear constraints input is in equation or Matrix form For a complete list of commands and other Optimization package information refer to the Optimization help page 5 6 Statistics 193 5 6 Statistics The Statistics package provides tools for mathematical statistics and data analysis The package supports a wide range of common statistical tasks including quantitative and graphical data analysis simulation and curve fitting In addition to standard data analysis tools the Statistics package provides a wide range of symbolic and numeric tools for computing with random variables The package supports over 35 major probability distributions and can be extended to include new distributions Probability Distributions and Random Variables The Statistics package supports e Continuous distributio
351. x x 0 Error does not have a taylor expansion try series To generate a truncated series expansion of a function about a point use the series command gt series Ci x x 0 y n 4 6 O x By default Maple performs series calculations up to order 6 To use a different order specify a non negative integer third argument gt expansion Sseries Ci t t 0 4 expansion Y In t f4 o 7 To set the order for all computations use the Order environment variable For information about the Order variable and the O 7 term refer to the Order help page 5 4 Calculus 183 The expansion 1s of type series Some commands for example plot do not accept arguments of type series To use the expansion you must convert it to a polynomial using the con vert polynom command l gt plot Ci t convert expansion polynom t 2 4 For information on Maple types and type conversions see Maple Expressions page 337 For information on plotting see Plots and Animations page 237 Integration Maple can perform symbolic and numeric integration To compute the indefinite integral of an expression Se l f dx 1 In the Expression palette click the indefinite integration item j 2 Specify the integrand and variable of integration and then evaluate it For example to integrate xsin ax with respect to x 184 5 Mathematical Problem Solving gt jx sin a x dx sin ax
352. x gt Circle lt 0 0 gt 1 output plot gt Linelnt VectorField lt y x gt Circle lt 0 0 gt 1 output integrat Qn sin r cos t 7 df 5 7 Ig To evaluate the integral returned by the output integral calling sequence use the value command gt value 5 7 27 5 8 By default the LineInt command returns the value of the integral gt Linelnt VectorField lt y x x y gt Circle lt 0 0 gt r 207 For more information on the Student package refer to the Student help page 206 5 Mathematical Problem Solving Calculus Problem Solving Examples Maple is a powerful application with many resources to guide you The following examples provide you with scenarios to learn about using Maple resources and the Maple program When using Maple to solve a problem consider the following process 1 Formulate your problem 2 Obtain Maple resources that allow you to solve it Problem Scenario A Your company is designing a bottle for its new spring water product The bottle must contain 18 ounces of water and the height is fixed The design includes an undulating curved surface You know the amplitude and equation of the curve but you must find the radius You require the Volume of Re volution Scenario B You want to teach your students the concept of a Volume of Revolution Specifically you want to plot and compute the volume of a solid generated by rotating
353. x y 20 x iS 2 x gt f S 7 2 1 4032680522 For more information on defining and using functions see Functional Operators page 343 Other Specialized Solvers In addition to equations and inequations Maple can solve other equations including e Ordinary differential equations ODEs e Partial differential equations PDEs e Integer equations e Integer equations in a finite field e Linear systems e Recurrence relations Ordinary Differential Equations ODEs Maple can solve ODEs and ODE systems including initial value and boundary value problems symbolically and numerically ODE Analyzer Assistant The ODE Analyzer Assistant is a point and click interface to the Maple ODE solving routines To open the ODE Analyzer e From the Tools menu select Assistants and then ODE Analyzer Maple inserts the dsolve interactive calling sequence in the document The ODE Analyzer Assistant Figure 4 3 is displayed 122 4 Basic Computations Ed ODE Analyzer Assistant Differential Equations Conditions Parameters A C 1 E m Solve Numerically Salve Symbolically Classify Figure 4 3 ODE Analyzer Assistant In the main ODE Analyzer Assistant window you can define ODESs initial or boundary value conditions and parameters To define derivatives use the diff command For example a x t dr diff x t t corresponds to Ul and diff x t t t corresponds to For more dt information on the
354. xample meters and grams and over 550 units with various contexts for example standard miles and U S survey miles Maple also contains two Units palettes that allow you to enter the unit for an expression quickly The Vector Calculus package is a collection of commands that perform multivariate and vector calculus operations A large set of predefined or thogonal coordinate systems 1s available All computations in the package can be performed in any of these coordinate systems It contains a facility for adding a custom but orthogonal coordinate system and using that new coordinate system for your computations Palettes are collections of related items that you can insert by clicking or dragging For ex ample see Figure 3 1 88 e 3 Worksheet Mode Expression yas f f dx y I j f lim f a b a b ah oe a E h a a ja Ja it lal e Infa log log a sin a cos a tan x fla Fla b f a oy f i fi x YA a h z f x la x x gt a i Figure 3 1 Expression Palette You can use palettes to enter input h fdx For example evaluate a definite integral using the definite integration item in the Expression palette In 2 D Math clicking the definite integration item inserts 1 Enter values in the placeholders To move to the next placeholder press Tab Note If pressing the Tab key inserts a tab click the Tab icon in the toolbar 2 To evaluate the integral press Enter l ta
355. y x consists of the right half of the line on the line y x with the left half of the line y x as shown x x2a 5 Sd Sg 2 1 0 1 2 3 4 5 I EA Memory 0 37M Time 0 03s Math Mode Ready Figure 1 1 The Maple Environment Starting the Standard Document Interface To start Maple on From the Start menu select All Programs Maple 14 Maple 14 Alternatively Double click the Maple 14 desktop icon Macintosh 1 From the Finder select Applications and Maple 14 2 Double click Maple 14 4 e 1 Getting Started Enter the full path for example usr local maple bin xmaple Alternatively 1 Add the Maple directory for example usr local maple bin to your command search path 2 Enter xmaple The first Maple session opens with a Startup dialog explaining the difference between Document Mode and Worksheet Mode Using either mode you can create high quality in teractive mathematical documents Each mode offers the same features and functionality the only difference is the default input region of each mode Document Mode Document mode uses Document Blocks as the default input region to hide Maple syntax A Document Block region is indicated by two triangles located in the vertical Markers column along the left pane of the Maple Document 2 lt L If the Markers column is not visible open the View menu and select Markers This allows you to focus on the problem instead of the commands used to
356. y the animation back ward Single iz Single Run the animation in single Ee A cycle mode The animation is dis Continuous played only once Continuous Run the animation in continuous mode The animation re peats until you stop it Set the animation to play at a faster or slower speed Determine the coordinates of a 2 D plot at the position of the cursor Frames per Eps second om Zoom into or out of the plot by chan ging the view ranges 10 k amp Pan ah Pan the plot by changing the view ranges Rotate 3 D di Rotate a three dimensional plot to see it from a different point of view You can also run the animation using the context menu or the Plot menu 6 7 Customizing Animations The display options that are available for static plots are also available for Maple animations Interactive Plot Builder Animation Options Using the Interactive Plot Builder you can apply various plot options within the Plot Options window See Interactive Plot Builder page 270 6 7 Customizing Animations 277 Context Menu Options As with static plots you can apply plot options to the animation by right clicking Con trol click for Macintosh the animation output Customize the animation using the context menu 1 To change the line style right click the plot region Select Style Point 2 To remove the axes select Axes None The animate Command Options The animate command offers a
357. y to perform data analysis For more information refer to the Statistics InteractiveDataAnalysis help page 5 7 Teaching and Learning with Maple Table 5 10 lists the available resources for instructors and students For additional resources see Available Resources page 58 Table 5 10 Student and Instructor Resources Resource Description Student Packages and Tutors The Student package contains computational and visualization plotting and animation functionality and point and click inter faces for explaining and exploring concepts Tools Tutors For more information refer to the Student help page Maple Portal The Maple Portal includes material designed for all Maple users as well as specific portals for students and educators The Maple Portal includes e How Dol topics that give quick answers to essential ques tions Tutorials that provide an overview of topics from getting started to plotting and working with matrices Navigation to portals with specialized information for stu dents math educators and engineers Access the portal from the Help menu Help Manuals Re sources and More Maple Portal Mathematics and Engineering Dic The Maple Help System has an integrated dictionary of over tionary 5000 mathematics and engineering terms You can search the dictionary by entering a term in the Help System search field Maple Application Center The Maple Application Center contains tutorials and appl
358. y values in the placeholders 3 To execute the operation and display the result press Ctrl Command for Macintosh or Enter 2 For example to evaluate re inline 70 2 Document Mode 1 Using the Expression palette enter the partial derivative See Example 1 Enter a Partial Derivative page 65 2 Press Ctrl Command for Macintosh 2 e 2fe Of Context Menus A context menu 1s a pop up menu that lists the operations and actions you can perform on a particular expression See Figure 2 1 2 11 9 T Copy as MathML Paste Ctrl Evaluate Evaluate and Display Inline Ctrl Explore Apply a Command Approximate Assign to a Mame Denominator Numerator Integer Functions b Units b 2 D Math I Figure 2 1 Context Menu To display the context menu for an expression e Right click Control click for Macintosh the expression The context menu is displayed beside the mouse pointer You can evaluate expressions using context menus e The Evaluate and Display Inline operation see Figure 2 1 is equivalent to pressing Ctrl Command for Macintosh That is it inserts an equal sign and then the value of the expression 2 6 Performing Computations 71 e The Evaluate operation see Figure 2 1 is equivalent to pressing Enter That is it eval uates the expression and displays the result centered on the following line For more information on evaluation
359. you have chosen This is for Maple procedures and other code that is best input in 1 D Math For more information see the helpnavigator help page Copying Examples Instead of opening the entire page as a document you can copy the Examples section only To copy examples 1 With the help page displayed in the right pane of the help system from the Edit menu select Copy Examples 2 Close or minimize the Help Navigator and return to your document 3 In your document place the cursor at the location where you want to paste the examples 4 From the Edit menu select Paste The Examples section of the help page is inserted as executable content in your document 58 1 Getting Started 1 6 Available Resources Your work with Maple is supported by numerous resources Resources Available through the Maple Help System Help Pages Use the help system to find information about a specific topic command package or feature For more information see The Maple Help System page 54 Dictionary More than 5000 mathematical and engineering terms with over 300 figures and plots 1 From the Help menu select Maple Help 2 Enter a search term Dictionary entries that match your query are displayed in the left pane with a b icon Tutorials and the Maple Portal The Maple Portal includes material designed for all Maple users from new users to users who want more advanced tutorials The Maple Portal also includes specific sections
360. you use Maple to analyze this expression and replace this form with a representation that is more accurate for small values of x a fully accurate 10 digit result can be obtained 106 4 Basic Computations gt taylor x sin x x ot Bs dhe al is 120 X O x x 0 00001 1 666666667 107 For information on evaluating an expression at a point see Substituting a Value fora Subexpression page 359 For information on creating a series approximation see Series page 182 For more information on floating point numbers refer to the float and type float help pages 4 3 Integer Operations In addition to the basic arithmetic operators Maple has many specialized commands for performing more complicated integer computations such as factoring an integer testing whether an integer is a prime number and determining the greatest common divisor GCD of a pair of integers Note Many integer operations are available as task templates Tools Tasks Browse under Integer Operations You can quickly perform many integer operations using context menus Selecting an integer and then right clicking for Macintosh Control clicking displays a context menu with in teger commands For example the context menu item Integer Factors applies the ifactor command to compute the prime factorization of the given integer See Figure 4 1 4 3 Integer Operations 107 9469629 9469629 Copy as MathML Numeric For
361. yscale 6 3 Customizing Plots 267 Change the axes style 4 Select Axes Boxed Alter the glossiness 5 Select Glossiness and then select Set Using the slider adjust the level of glossiness The plot and plot3d Options If you are using commands to insert a plot you can specify plot options as arguments at the end of the calling sequence You can specify the options in any order Applying plot options in the command syntax offers a few more options and greater control than what is available in the Interactive Plot Builder and context menus Table 6 3 Common Plot Options lightmodel 3 D Controls the light model to illuminate the plot one of none light1 light2 light3 or light4 linestyle Defines the dash pattern used to render lines in the plot one of dot dash dashdot longdash solid spacedash and spacedot legend 2 D Defines a legend for the plot Controls the minimum total number of points generated Controls the scaling of the graph one of constrained or unconstrained shading 3 D Defines how the surface is colored one of xyz xy z zgrayscale zhue or none style Defines how the surface is to be drawn one of line point polygon or polygonoutline for 2 D plots contour point surface surfacecontour surfacewireframe wireframe or wireframeopaque for 3 D plots symbol Defines the symbol for points in the plot one of asterisk box circle cross diagonalcross diamond point solid

Maple User Manual - University Information & Technology Services

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Maple User Manual - University Information &amp; Technology Services

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Maple User Manual - University Information & Technology Services