command the brilliance of a thousand mathematicians Waterloo Maple Inc. 57 Erb Street West Waterloo, Ontario | Canada N2L 6C2 Learning Guide tel: 1.519.747.2373 | fax: 1.519.747.5284 [email protected] | www.maplesoft.com North American Sales: 1.800.267.6583 Learning Guide © 2002 Waterloo Maple Inc. Maple is a registered trademark of Waterloo Maple Inc. M-0028-00-E Printed in Canada Maple 8 Learning Guide Based in part on the work of B. W. Char c 2002 by Waterloo Maple Inc. ­ ii ¯ Waterloo Maple Inc. 57 Erb Street West Waterloo, ON N2L 6C2 Canada Maple and Maple V are registered trademarks of Waterloo Maple Inc. Maplets is a trademark of Waterloo Maple Inc. c 2002, 2001, 2000, 1998, 1996 by Waterloo Maple Inc. All rights ­ reserved. The electronic version (PDF) of this book may be downloaded and printed for personal use or stored as a copy on a personal machine. The electronic version (PDF) of this book may not be distributed. 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This document was produced using a special version of Maple that reads and updates LATEX files. Printed in Canada ISBN 1-894511-26-3 Contents 1 Introduction to Maple 1 1.1 Manual Set . 3 2 Mathematics with Maple: the Basics 5 2.1 Introduction . 5 2.2 Numerical Computations . 7 Integer Computations . 7 Exact Arithmetic—Rationals, Irrationals, and Constants . 8 Floating-Point Approximations . 11 Arithmetic with Special Numbers . 13 Mathematical Functions . 14 2.3 Basic Symbolic Computations . 16 2.4 Assigning Expressions to Names . 17 2.5 Basic Types of Maple Objects . 19 Expression Sequences . 19 Lists . 21 Sets . 22 Operations on Sets and Lists . 24 Arrays . 25 Tables . 29 Strings . 30 2.6 Expression Manipulation . 31 The simplify Command . 31 The factor Command . 33 The expand Command . 33 The convert Command . 34 The normal Command . 35 The combine Command . 36 The map Command . 36 The lhs and rhs Commands . 38 iii iv Contents ¯ The numer and denom Commands . 38 The nops and op Commands . 38 Common Questions about Expression Manipulation . 39 2.7 Conclusion . 41 3 Finding Solutions 43 3.1 Simple solve ......................... 43 Verifying Solutions . 45 Restricting Solutions . 47 Exploring Solutions . 48 The unapply Command . 49 The assign Command . 51 The RootOf Command . 52 3.2 Solving Numerically: fsolve . 53 Limitations on solve ..................... 55 3.3 Other Solvers . 57 Finding Integer Solutions . 57 Finding Solutions Modulo m . 58 Solving Recurrence Relations . 58 3.4 Polynomials . 58 Sorting and Collecting . 59 Mathematical Operations . 61 Coeácients and Degrees . 62 Root Finding and Factorization . 62 3.5 Calculus . 64 3.6 Differential Equations: dsolve . 70 3.7 The Organization of Maple . 76 3.8 The Maple Packages . 78 List of Packages . 78 The Student Calculus1 Package (Single Variable) . 83 The LinearAlgebra Package . 88 The Matlab Package . 90 The Statistics Package . 91 The Linear Optimization Package . 94 3.9 Conclusion . 96 4 Graphics 97 4.1 Graphing in Two Dimensions . 97 Parametric Plots . 99 Polar Coordinates . 101 Functions with Discontinuities . 104 Contents v ¯ Multiple Functions . 107 Plotting Data Points . 109 Refining Plots . 111 4.2 Graphing in Three Dimensions . 112 Parametric Plots . 114 Spherical Coordinates . 114 Cylindrical Coordinates . 117 Refining Plots . 118 Shading and Lighting Schemes . 119 4.3 Animation . 120 Animation in Two Dimensions . 121 Animation in Three Dimensions . 123 4.4 Annotating Plots . 124 4.5 Composite Plots . 127 Placing Text in Plots . 129 4.6 Special Types of Plots . 130 4.7 Manipulating Graphical Objects . 135 4.8 Code for Color Plates . 140 4.9 Conclusion . 143 5 Evaluation and Simplification 145 5.1 Mathematical Manipulations . 145 Expanding Polynomials as Sums . 146 Collecting the Coeácients of Like Powers . 148 Factoring Polynomials and Rational Functions . 150 Removing Rational Exponents . 153 Combining Terms . 154 Factored Normal Form . 155 Simplifying Expressions . 157 Simplification with Assumptions . 158 Simplification with Side Relations . 159 Sorting Algebraic Expressions . 160 Converting Between Equivalent Forms . 162 5.2 Assumptions . 163 The assume Facility . 163 The assuming Command . 168 5.3 Structural Manipulations . 169 Mapping a Function onto a List or Set . 169 Choosing Elements from a List or Set . 172 Merging Two Lists . 173 Sorting Lists . 174 vi Contents ¯ The Parts of an Expression . 177 Substitution . 185 Changing the Type of an Expression . 189 5.4 Evaluation Rules . 191 Levels of Evaluation . 191 Last-Name Evaluation . 192 One-Level Evaluation . 195 Commands with Special Evaluation Rules . 196 Quotation and Unevaluation . 197 Using Quoted Variables as Function Arguments . 200 Concatenation of Names . 201 5.5 Conclusion . 203 6 Examples from Calculus 205 6.1 Introductory Calculus . 205 The Derivative . 205 A Taylor Approximation . 211 The Integral . 223 Mixed Partial Derivatives . 227 6.2 Ordinary Differential Equations . 231 The dsolve Command . 232 Example: Taylor Series . 247 When You Cannot Find a Closed Form Solution . 251 Plotting Ordinary Differential Equations . 252 Discontinuous Forcing Functions . 256 6.3 Partial Differential Equations . 261 The pdsolve Command . 261 Changing the Dependent Variable in a PDE . 263 Plotting Partial Differential Equations . 265 6.4 Conclusion . 267 7 Input and Output 269 7.1 Reading Files . 269 Reading Columns of Numbers from a File . 270 Reading Commands from a File . 272 7.2 Writing Data to a File . 273 Writing Columns of Numerical Data to a File . 273 Saving Expressions in Maple's Internal Format . 275 Converting to LATEX Format . 276 7.3 Exporting Whole Worksheets . 278 Plain Text . 278 Contents vii ¯ Maple Text . 278 LATEX ............................. 279 HTML and HTML with MathML . ..