Maple 12 Tutorial - University Of Texas At Austin
Maple 12 TutorialAugust 2012
Maple 12 TutorialTable of ContentsSECTION 1. INTRODUCTION . 3ABOUT THE DOCUMENT . 3MAPLE INTRODUCTION . 3ACCESS TO MAPLE . 3GETTING STARTED . 4THE MAPLE APPLICATION INTERFACE. 5THE HELP WINDOW. 8BASIC CONVENTIONS IN MAPLE . 9BASIC DATA STRUCTURES IN MAPLE . 11SECTION 3. NUMERICAL COMPUTATION . 12FLOATING POINT NUMBER COMPUTATION . 12INTEGER COMPUTATIONS . 13COMPLEX NUMBER COMPUTATION . 13BUILT -IN FUNCTIONS AND CONSTANTS . 14SECTION 4. SYMBOLIC COMPUTATION . 15SYMBOLIC COMPUTATIONS ON EXPRESSIONS . 15EQUATIONS . 16FUNCTIONS . 17SECTION 5. GRAPHICS . 18TWO-DIMENSIONAL PLOTS . 18THREE-DIMENSIONAL PLOTS . 21MULTIPLE PLOT DISPLAY . 23ANIMATION . 25SECTION 6. CALCULUS . 27DIFFERENTIATIONS . 27INTEGRALS . 31DIFFERENTIAL EQUATIONS . 32LIMIT FUNCTION . 34SECTION 7. LINEAR ALGEBRA . 34VECTORS . 34MATRICES . 37EIGENVECTORS AND EIGENVALUES. 42SECTION 8. CONCLUSION. 442The Division of Statistics Scientific Computation, The University of Texas at Austin
Maple 12 TutorialSection 1. IntroductionAbout The DocumentThis document introduces you to Maple 12. It is designed for beginning users of theMaple application. In order to make use of this document, you should have a basicknowledge of calculus, linear algebra, and differential equations. The document isorganized into 8 sections. The first section provides a brief introduction to Maple. Thesecond section gives an overview of the Maple Windows Development Environment andits basic conventions. The third and fourth sections explain how to use Maple to donumerical and symbolic computations. The fifth section covers the graphics functionalityof Maple, including 2D plotting, 3D plotting and animation. The sixth section shows howto use Maple to solve Calculus problems. The seventh section introduces linear algebra inMaple. The eighth section concludes the document with information about where to getfurther help and resources about Maple.Maple IntroductionMaple is a mathematical software package for symbolic computation. Conventionalmathematical software packages usually require numerical values for all variables. Incontrast, Maple can evaluate both symbolic and numerical expressions.Maple was first developed at the University of Waterloo, Canada in 1980. The objectivewas to design an algebra software for the researchers and students in Mathematics,Engineering, and Science. It was initially implemented in B on a Honeywell computerand soon was switched to C, one of the most popular programming languages.In 1988 Waterloo Maple Software was founded to directly sell and further improve theMaple system. Currently, the version formally supported by ITS and SSC is Maple 12.Maple is a computer software program for advanced mathematics. It is available for awide variety of platforms including Windows, UNIX and Linux. It includes utilities foralgebra, calculus, discrete mathematics, graphics, numerical calculation, and many otherareas of mathematics. It also provides a unique environment for rapid development ofmathematical programs using its vast library of built-in functions and operations.Section 2. An Overview of MapleAccess to MapleInformation Technology Services (ITS) at UT-Austin has discontinued its support ofMaple on the computers in the Student Microcomputer Facility (SMF); however,3The Division of Statistics Scientific Computation, The University of Texas at Austin
Maple 12 TutorialMaple 10 is installed on the ITS UNIX time sharing system, uts.cc.utexas.edu, running onthe Solaris 8 operating system. Maple 12 is not compatible with that vintage operatingsystem. A Windows version of Maple 12 is installed, however, on the ITS Windowsterminal server, wts.austin.utexas.edu. To access these installations, an ITS-issuedaccount must have UNIX service (uts), or Windows Terminal Services validation(austin). These mainframe validations require a minimum of 5000K disk storage, costing 10 per year.Maple licensing through UT Software Distribution Services has been discontinued.Licenses can be obtained directly from the vendor. Pricing and other information aboutMaple is available at:http://www.maplesoft.com/Getting StartedThe Maple launch method varies for different operating systems: On a Macintosh: Start Maple by double-clicking on the Maple icon. Maple willstart up with an empty worksheet. To begin using Maple, type commands atMaple's prompt in the command window. Alternatively you can start Maple bydouble-clicking on a previously saved Maple worksheet, in which case Maple willautomatically open that worksheet when it starts up.On a PC running Windows: Start Maple by double-clicking on the Maple icon.This will start up Maple with a blank worksheet. You can then begin typingMaple commands at the Maple worksheet prompt.On a Unix machine: The Maple executables are not in the default commandsearch path. You can add this directory manually to your path. If you are using theC-shell, type:setenv PATH /usr/local/maple/bin: PATHIf you are using the Bourne shell, type:DISPLAY /usr/local/maple/bin: PATH; export PATHThe command above can be put at the end of your .cshrc (or .profile) so that itwill be executed automatically. (Note: issue this command only after you have setyour environment variables such as PATH and MANPATH.)To start the command line implementation of Maple, type:maple4The Division of Statistics Scientific Computation, The University of Texas at Austin
Maple 12 TutorialAn X based version of Maple is also available. To run the X based versionof Maple you must first allow the system on which you are running Mapleto connect to your workstation or X-terminal. To do this, type:xhost Unix machine nameon your local workstation. Next, set the DISPLAY environment variableto point to your workstation or X-terminal. If you are using the C-shell,type:setenv DISPLAY your local machine name:0If you are using the Bourne shell, use:DISPLAY your local machine name:0; export DISPLAYthen you may run the X version of Maple by executing the command:xmaple &When Maple begins running, it will automatically execute the commandsin the file .mapleinit in your home directory, if it exists.The Maple Application InterfaceMaple has two types of display interfaces, one for simple command line input a Standardinterface for graphical display. The command line interface is used for simple ascii basedterminals such as vt100. The graphical interface requires a Windows or xtermenvironment. When using the Standard interface there are two modes, a document modethat hides commands by default and a worksheet mode that displays commands bydefault. As of the Maple 12 version the document mode is the default. Previously thedefault had been the worksheet mode. These two modes have the exact samefunctionality, just the input is different. Document mode input starts at a vertical slashprompt ( / ) and worksheet mode input starts at a red font right angle bracket prompt ( ).When Maple 12 is started, we see a window similar to that shown in Figure 1.This isMaple's document mode display for the Standard interface, the default upon launch.5The Division of Statistics Scientific Computation, The University of Texas at Austin
Maple 12 TutorialFigure 1. The Maple Window for Document ModeSometimes, however, it is more convenient use worksheet mode. You can launch a newMaple worksheet by selecting New Worksheet Mode from the File menu on thenavigation bar.Figure 2. The Maple Window for Worksheet Mode6The Division of Statistics Scientific Computation, The University of Texas at Austin
Maple 12 TutorialIn a worksheet Maple input is displayed in black type, with a Courier font, and text inputis italic. The Maple numeric or symbolic output is displayed in blue type in standard mathnotation, and a simple plot line is displayed in red.Figure 3. The Maple WindowSince the worksheet mode does not hide text it is probably an easier mode to use whenfirst learning to use Maple, so for this tutorial we will henceforth give the pictorialdisplays as worksheets, remembering the operational manipulations in document modework exactly the same. At the top of a Standard Maple interface, as is shown in Figure 3,there is a menu bar. It provides different options for you to save or edit the Maplesession. For example, clicking on the File menu and selecting Save will allow you tosave your current worksheet for later use; selecting Open from the same menu allows youto load an existing worksheet. Below the menu bar is a collection of buttons, which areshort-cuts for items found on the various menus. Clicking on one of these is muchquicker than searching for the command on the menu. For example, the button that lookslike a floppy disk ( ) saves the current worksheet. Below the shortcut buttons is the7The Division of Statistics Scientific Computation, The University of Texas at Austin
Maple 12 Tutorialcontext bar, which changes depending on what kind of item you currently have selected.For example, when graphics are selected there are buttons to adjust the graphics styles.The Help WindowMaple 12 has an extensive online help system to help you understand its language,procedures, and syntax. Each help page has a set of examples at the end. You can test theMaple commands by copying an example from a help page and pasting into theworksheet window.There are a few ways to get into the help system. One obvious one is to select from theHelp menu. If you are a new user, you may wish to choose Help Take a Tour of Maplewhich provides information to help you get started. After selecting Help Maple Help,there is search area from which information can be found with topic or text keys. Besidesthe help menu approach, there is another quick way to find help information. If youknow the help topic in advance, you can issue a Maple help command with the ?syntaxdirectly from a Maple worksheet. For example, ?solve will show you the solve() functionhelp information.The help window has two panes. The left pane contains the Help Navigator, while theright pane displays the help page (see Figure 4 for an example). In the help navigator, afolder icon indicates that a topic can be expanded into subtopics. Clicking a topicpreceded by a dot icon displays the associated help page. You can use this to browse forcommands you might not otherwise have known about.Figure 4. Example from Maple Help8The Division of Statistics Scientific Computation, The University of Texas at Austin
Maple 12 TutorialBasic Conventions in MapleEnter A CommandUsing Maple to enter and execute a command is very simple: merely input the commandat the Maple prompt (in Maple 12 an initially empty small rectangle with gray dashboundaries that expand as the expression is type in). When you press "Enter", Maple willexecute the input and the result will be shown on the line below, in the center of thescreen. The following example shows a simple numerical calculation. After you type1 2; and hit the return key, the result 3 will be shown in the center of screen.Example: Entering and executing a commandArithmetic Operators of MapleThe arithmetic operators of Maple are defined as follows in Table 1:Table ractionMultiplicationDivisionExponentiation */ or **1 2x-y54*4x/y2 3 or 2**3The precedence order of the Maple arithmetic operators is the same with the order ofmathematical arithmetic operators. Parentheses could be used for grouping to manipulatethe operational order. For example, (89-9)*2 first calculates 89-9, then performs themultiplication (89-9)*2. Without the parentheses, 89-9*2 will first calculate 9*2, then dothe subtraction 89-9*2.Referring to Previous Results and Adding CommentsIn Maple, percentage signs are used to refer to previously computed expressions.Specifically, the following operators are defined:9The Division of Statistics Scientific Computation, The University of Texas at Austin
Maple 12 Tutorial%Most recent%%Second most recent%%% Third most recentThe example below shows how we can use the percentage signs to simplify numericalcalculations. For each step, we also insert comments using the pound sign # : allcharacters that follow the pound sign # are considered as comments.Example: Using percentage signs to refer to previous computed resultsDefining An Expression Using : Most calculations in Maple will involve defining expressions. An expression is acombination of number(s), variable(s) and/or mathematical operator(s). To define anexpression, you assign it a name using the convention name: expression. In this way,whenever you want to use the expression, we can directly use the name without retypingthe expression. The example below assigns the name, expression1, to the expressionx 2 x 2.Example: Defining the expression x 2 x 2 expression1 : 10The Division of Statistics Scientific Computation, The University of Texas at Austin
Maple 12 TutorialHow to Handle MistakesIf you accidentally type incorrect syntax in the command line and hit enter, Maple willrespond with a syntax error. There are two ways to fix your error. You can go back,change the Maple command, and re-execute it; when Maple re-executes the command,the result will change. Or you can type restart at a prompt to clear the old informationand then type in the correct Maple commands.Basic Data Structures In MapleThe most fundamental data structures in Maple are expression sequences, lists and sets.They are commonly used as parameters in Maple commands.Expression SequencesAn expression sequence is one or more expressions separated by commas. The simpleway to create an expression sequence is to enter it directly. Enter the expression sequenceat the prompt. Separate each element in the sequence with commasExample: Creating an expression sequence Alternatively, the seq() function can be used to create an expression sequence. It operatesvery quickly and improves the efficiency of your Maple program. In order to generate thesequences f(m), f(m 1), f(m 2),. f(n), you would type: seq(f(i),i m.n) # f represents an expression with the index of iIn the following example, f is i2. We use the seq() function to create the expressionsequence f(2), f(3) and f(4).Example: Creating the expression sequence of the square of i using seq() function 11The Division of Statistics Scientific Computation, The University of Texas at Austin
Maple 12 TutorialLists and SetsLists and Sets are usually used as the arguments of functions. Expression sequences areused within Lists and Sets. A list is an expression sequence enclosed in square brackets.It preserves the order and repetition of elements in a list. In the following example, wecreate a list, [1,1,3,5,5] by enclosing the expression sequence 1,1,3,5,5, in squarebrackets. We then name it as list1. We can see list1 preserves the list order and repeatedelements.Example: Defining a List A set is an expression sequence enclosed in curly braces. It doesn't preserve order orrepetition of elements. Thus, a Maple set has the same properties of a Mathematical set.For example, we define set1 and set2 with different expression sequences inputs.However the loss of order and repetition results in identical sets.Example: Defining a Set Section 3. Numerical ComputationMaple can do floating point, integer and complex numerical computations.Floating Point Number ComputationOne of the important Maple properties is the ability to do exact arithmetic: rationalnumbers are not reduced to floating point approximation. In this way, round-off error canbe avoided.12The Division of Statistics Scientific Computation, The University of Texas at Austin
Maple 12 TutorialExample: Computation Without Round-off Error If an approximate result is needed, the evalf function can be used to convert an exactnumerical result to an approximate floating point number result.Example: Converting an exact numerical result to a floating point number 2.325000000Or you can include float number(s) in an expression and it will automatically causeMaple to produce an approximate floating point result.Example: Using a floating point number inclusion to create floating point number result 2.325000000Integer ComputationsMaple can work with arbitrarily large integers. If a number is too big and spans morethan one line on the screen, a backslash can be used to indicate the number is stillcontinuous.Example: Working with large integers. 00Complex Number ComputationWhen Maple does complex number calculations, a capital I is used to represent thesquare root of minus one.13The Division of Statistics Scientific Computation, The University of Texas at Austin
Maple 12 TutorialExample: Built -In Functions and ConstantsMaple has many built-in standard functions and constants, which can be applied whendoing numerical calculations. The commonly used numerical functions and constants areshown in the following Table 2 and Table 3.Table 2. Selected Built-In Mathematical
Maple 12 Tutorial 4 The Division of Statistics Scientific Computation, The University of Texas at Austin Maple 10 is installed on the ITS UNIX time sharing system, uts.cc.utexas.edu, running on the Solaris 8 operating system. Maple
We will touch on all of these capabilities in this tutorial. Maple worksheets This document you are reading is called a Maple worksheet; it combines text with Maple commands and their results, including graphics. (Here I assume that you are reading this file in Maple , not as a printed document. . multiplication, and "/" for division.
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2019 installation: 13. Review the Pre-Installation Summary: 14.Maple 2019 will install on your computer: 15. Decide if you would like to activate Maple 2019 now: 16. Input your purchase code and then the necessary information to activate: . Maple 2019 Installation Guide
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