Mathematica by Example, Fourth Edition

577 Pages·2008·13.06 MB·English

Mathematica by
Example
This page intentionally left blank
Mathematica by
Example
Fourth Edition
Martha L. Abell and James P. Braselton
Department of Mathematical Sciences
Georgia Southern University
Statesboro, Georgia
AMSTERDAM • BOSTON • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
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09 10 11 12 9 8 7 6 5 4 3 2 1
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
CHAPTER 1 Getting Started 1
1.1 Introduction to Mathematica . . . . . . . . . . . . . . . . . . . . . 1
A Note Regarding Different Versions of
Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Getting Started with Mathematica . . . . . . . . . . . . . . 3
Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Five Basic Rules of Mathematica Syntax . . . . . . . . . . . . . . 13
1.2 Loading Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 Packages Included with Older Versions of
Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2 Loading New Packages . . . . . . . . . . . . . . . . . . . . 15
1.3 Getting Help from Mathematica . . . . . . . . . . . . . . . . . . . 17
Mathematica Help . . . . . . . . . . . . . . . . . . . . . . 24
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
CHAPTER 2 Basic Operations on Numbers,
Expressions, and Functions 31
2.1 Numerical Calculations and Built-in Functions . . . . . . . . . . . 31
2.1.1 Numerical Calculations . . . . . . . . . . . . . . . . . . . . 31
2.1.2 Built-in Constants . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.3 Built-in Functions . . . . . . . . . . . . . . . . . . . . . . . 35
A Word of Caution . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Expressions and Functions: Elementary Algebra . . . . . . . . . . 39
2.2.1 Basic Algebraic Operations on Expressions . . . . . . . . 39
2.2.2 Naming and Evaluating Expressions . . . . . . . . . . . . . 44
2.2.3 Defining and Evaluating Functions . . . . . . . . . . . . . 47
2.3 Graphing Functions, Expressions, and Equations . . . . . . . . . 52
2.3.1 Functions of a Single Variable . . . . . . . . . . . . . . . . 52
2.3.2 Parametric and Polar Plots in Two Dimensions . . . . . . 65
2.3.3 Three-Dimensional and Contour Plots:
Graphing Equations . . . . . . . . . . . . . . . . . . . . . . 71
2.3.4 Parametric Curves and Surfaces in Space . . . . . . . . . . 82
2.3.5 Miscellaneous Comments . . . . . . . . . . . . . . . . . . . 94
2.4 Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.4.1 Exact Solutions of Equations . . . . . . . . . . . . . . . . . 100
2.4.2 Approximate Solutions of Equations . . . . . . . . . . . . 110
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 v
vi Contents
CHAPTER 3 Calculus 117
3.1 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.1.1 Using Graphs and Tables to Predict Limits . . . . . . . . . 117
3.1.2 Computing Limits . . . . . . . . . . . . . . . . . . . . . . . 121
3.1.3 One-Sided Limits . . . . . . . . . . . . . . . . . . . . . . . . 123
3.1.4 Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.2 Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.2.1 Definition of the Derivative . . . . . . . . . . . . . . . . . 128
3.2.2 Calculating Derivatives . . . . . . . . . . . . . . . . . . . . 135
3.2.3 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . 138
3.2.4 Tangent Lines. . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.2.5 The First Derivative Test and Second
Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.2.6 Applied Max/Min Problems. . . . . . . . . . . . . . . . . . 156
3.2.7 Antidifferentiation . . . . . . . . . . . . . . . . . . . . . . . 164
3.3 Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
3.3.1 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
3.3.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . 174
3.3.3 Approximating Definite Integrals . . . . . . . . . . . . . . 179
3.3.4 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
3.3.5 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
3.3.6 Solids of Revolution . . . . . . . . . . . . . . . . . . . . . . 190
3.4 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
3.4.1 Introduction to Sequences and Series. . . . . . . . . . . . 201
3.4.2 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . 205
3.4.3 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . 209
3.4.4 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . 210
3.4.5 Taylor and Maclaurin Series . . . . . . . . . . . . . . . . . 213
3.4.6 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 217
3.4.7 Other Series . . . . . . . . . . . . . . . . . . . . . . . . . . 220
3.5 Multivariable Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 221
3.5.1 Limits of Functions of Two Variables . . . . . . . . . . . . 222
3.5.2 Partial and Directional Derivatives . . . . . . . . . . . . . . 224
3.5.3 Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . 238
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
CHAPTER 4 Introduction to Lists and Tables 251
4.1 Lists and List Operations . . . . . . . . . . . . . . . . . . . . . . . 251
4.1.1 Defining Lists . . . . . . . . . . . . . . . . . . . . . . . . . . 251
4.1.2 Plotting Lists of Points . . . . . . . . . . . . . . . . . . . . 258
4.2 Manipulating Lists: More on Part and Map . . . . . . . . . . . . . 269
4.2.1 More on Graphing Lists: Graphing Lists of Points
Using Graphics Primitives . . . . . . . . . . . . . . . . . . 277
4.2.2 Miscellaneous List Operations . . . . . . . . . . . . . . . . 283
Contents vii
4.3 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 283
4.3.1 Approximating Lists with Functions. . . . . . . . . . . . . 283
4.3.2 Introduction to Fourier Series . . . . . . . . . . . . . . . . 287
4.3.3 The Mandelbrot Set and Julia Sets . . . . . . . . . . . . . . 299
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
CHAPTER 5 Matrices and Vectors: Topics from
Linear Algebra and Vector Calculus 317
5.1 Nested Lists: Introduction to Matrices, Vectors, and
Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
5.1.1 Defining Nested Lists, Matrices, and Vectors. . . . . . . . 317
5.1.2 Extracting Elements of Matrices . . . . . . . . . . . . . . . 322
5.1.3 Basic Computations with Matrices . . . . . . . . . . . . . 325
5.1.4 Basic Computations with Vectors . . . . . . . . . . . . . . 329
5.2 Linear Systems of Equations . . . . . . . . . . . . . . . . . . . . . 337
5.2.1 Calculating Solutions of Linear Systems of
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
5.2.2 Gauss–Jordan Elimination . . . . . . . . . . . . . . . . . . . 342
5.3 Selected Topics from Linear Algebra . . . . . . . . . . . . . . . . 349
5.3.1 Fundamental Subspaces Associated with
Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
5.3.2 The Gram–Schmidt Process . . . . . . . . . . . . . . . . . 351
5.3.3 Linear Transformations . . . . . . . . . . . . . . . . . . . . 355
5.3.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . 358
5.3.5 Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . 361
5.3.6 The QR Method . . . . . . . . . . . . . . . . . . . . . . . . 364
5.4 Maxima and Minima Using Linear Programming. . . . . . . . . . 366
5.4.1 The Standard Form of a Linear Programming
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
5.4.2 The Dual Problem . . . . . . . . . . . . . . . . . . . . . . . 368
5.5 Selected Topics from Vector Calculus. . . . . . . . . . . . . . . . 374
5.5.1 Vector-Valued Functions . . . . . . . . . . . . . . . . . . . 374
5.5.2 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 384
5.5.3 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . 387
5.5.4 A Note on Nonorientability . . . . . . . . . . . . . . . . . 391
5.5.5 More on Tangents, Normals, and Curvature in R3 . . . . 404
5.6 Matrices and Graphics . . . . . . . . . . . . . . . . . . . . . . . . 415
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
CHAPTER 6 Applications Related to Ordinary and
Partial Differential Equations 435
6.1 First-Order Differential Equations . . . . . . . . . . . . . . . . . . 435
6.1.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . 435
6.1.2 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . 442
viii Contents
6.1.3 Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . 450
6.1.4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 453
6.2 Second-Order Linear Equations . . . . . . . . . . . . . . . . . . . 457
6.2.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 457
6.2.2 Constant Coefficients . . . . . . . . . . . . . . . . . . . . . 458
6.2.3 Undetermined Coefficients . . . . . . . . . . . . . . . . . . 464
6.2.4 Variation of Parameters . . . . . . . . . . . . . . . . . . . . 470
6.3 Higher-Order Linear Equations . . . . . . . . . . . . . . . . . . . . 472
6.3.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 472
6.3.2 Constant Coefficients . . . . . . . . . . . . . . . . . . . . . 473
6.3.3 Undetermined Coefficients . . . . . . . . . . . . . . . . . . 475
6.3.4 Laplace Transform Methods . . . . . . . . . . . . . . . . . 481
6.3.5 Nonlinear Higher-Order Equations. . . . . . . . . . . . . . 492
6.4 Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . 492
6.4.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 492
6.4.2 Nonhomogeneous Linear Systems . . . . . . . . . . . . . . 505
6.4.3 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . 511
6.5 Some Partial Differential Equations . . . . . . . . . . . . . . . . . 532
6.5.1 The One-Dimensional Wave Equation . . . . . . . . . . . . 532
6.5.2 The Two-Dimensional Wave Equation . . . . . . . . . . . . 537
6.5.3 Other Partial Differential Equations . . . . . . . . . . . . . 547
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
References 557
Index 559
Preface
Mathematica by Example bridges the gap that exists between the very
elementaryhandbooksavailableonMathematicaandthosereferencebooks
writtenfortheadvancedMathematicausers.Thisbookisanappropriateref-
erence for all users of Mathematica and, in particular, for beginning users
such as students, instructors, engineers, businesspeople, and other profes-
sionals first learning to use Mathematica. This book introduces the very
basiccommandsandincludestypicalexamplesofapplicationsofthesecom-
mands.Inaddition,thetextalsoincludescommandsusefulinareassuchas
calculus,linearalgebra,businessmathematics,ordinaryandpartialdifferen-
tialequations,andgraphics.Inallcases,however,examplesfollowtheintro-
duction of new commands. Readers from the most elementary to advanced
levelswillfindthattherangeoftopicscoveredaddressestheirneeds.
Taking advantage of Version 6 of Mathematica, Mathematica by Exam-
ple, Fourth Edition, introduces the fundamental concepts of Mathematica
to solve typical problems of interest to students, instructors, and scientists.
The fourth edition is an extensive revision of the text. Features that make
this edition easy to use as a reference and as useful as possible for the
beginner include the following:
1. Version 6 compatibility. All examples illustrated in this book were
completed using Version 6 of Mathematica. Although many com-
putations can continue to be carried out with earlier versions of
Mathematica,wehavetakenadvantageofthenewfeaturesinVersion
6 as much as possible.
2. Applications. New applications, many of which are documented by
references from a variety of fields, especially biology, physics, and
engineering, are included throughout the text.
3. Detailed table of contents. The table of contents includes all chap-
ter, section, and subsection headings. Along with the comprehensive
index, we hope that users will be able to locate information quickly
and easily.
4. Additional examples. We have considerably expanded the topics
throughout the book. The results should be more useful to instruc-
tors, students, businesspeople, engineers, and other professionals
using Mathematica on a variety of platforms. In addition, several
sections have been added to make it easier for the user to locate
information.
ix