Mathematica by Example (eBook)

Front Cover 1
Mathematica by Example 4
Copyright Page 5
Table of Contents 6
Preface 10
Chapter 1. Getting Started 14
1.1 Introduction to Mathematica 14
A Note Regarding Different Versions of Mathematica 15
1.1.1 Getting Started with Mathematica 16
Preview 26
Five Basic Rules of Mathematica Syntax 26
1.2 Loading Packages 26
1.2.1 Packages Included wi.th Older Versions of Mathematica 27
1.2.2 Loading New Packages 28
1.3 Getting Help from Mathematica 30
Mathematica Help 37
1.4 Exercises 41
Chapter 2. Basic Operations on Numbers, Expressions, and Functions 44
2.1 Numerical Calculations and Built-in Functions 44
2.1.1 Numerical Calculations 44
2.1.2 Built-in Constants 47
2.1.3 Built-in Functions 48
A Word of Caution 51
2.2 Expressions and Functions: Elementary Algebra 52
2.2.1 Basic Algebraic Operations on Expressions 52
2.2.2 Naming and Evaluating Expressions 57
2.2.3 Defining and Evaluating Functions 60
2.3 Graphing Functions, Expressions, and Equations 65
2.3.1 Functions of a Single Variable 65
2.3.2 Parametric and Polar Plots in Two Dimensions 78
2.3.3 Three-Dimensional and Contour Plots: Graphing Equations 84
2.3.4 Parametric Curves and Surfaces in Space 95
2.3.5 Miscellaneous Comments 107
2.4 Solving Equations 113
2.4.1 Exact Solutions of Equations 113
2.4.2 Approximate Solutions of Equations 123
2.5 Exercises 128
Chapter 3. Calculus 130
3.1 Limits and Continuity 130
3.1.1 Using Graphs and Tables to Predict Limits 130
3.1.2 Computing Limits 134
3.1.3 One-Sided Limits 136
3.1.4 Continuity 137
3.2 Differential Calculus 141
3.2.1 Definition of the Derivative 141
3.2.2 Calculating Derivatives 148
3.2.3 Implicit Differentiation 151
3.2.4 Tangent Lines 152
3.2.5 The First Derivative Test and Second Derivative Test 161
3.2.6 Applied Max/Min Problems 169
3.2.7 Antidifferentiation 177
3.3 Integral Calculus 181
3.3.1 Area 181
3.3.2 The Definite Integral 187
3.3.3 Approximating Definite Integrals 192
3.3.4 Area 193
3.3.5 Arc Length 199
3.3.6 Solids of Revolution 203
3.4 Series 214
3.4.1 Introduction to Sequences and Series 214
3.4.2 Convergence Tests 218
3.4.3 Alternating Series 222
3.4.4 Power Series 223
3.4.5 Taylor and Maclaurin Series 226
3.4.6 Taylor’s Theorem 230
3.4.7 Other Series 233
3.5 Multivariable Calculus 234
3.5.1 Limits of Functions of Two Variables 235
3.5.2 Partial and Directional Derivatives 237
3.5.3 Iterated Integrals 251
3.6 Exercises 259
Chapter 4. Introduction to Lists and Tables 264
4.1 Lists and List Operations 264
4.1.1 Defining Lists 264
4.1.2 Plotting Lists of Points 271
4.2 Manipulating Lists: More on Part and Map 282
4.2.1 More on Graphing Lists: Graphing Lists of Points Using Graphics Primitives 290
4.2.2 Miscellaneous List Operations 296
4.3 Other Applications 296
4.3.1 Approximating Lists with Functions 296
4.3.2 Introduction to Fourier Series 300
4.3.3 The Mandelbrot Set and Julia Sets 312
4.4 Exercises 324
Chapter 5. Matrices and Vectors: Topics from Linear Algebra and Vector Calculus 330
5.1 Nested Lists: Introduction to Matrices, Vectors, and Matrix Operations 330
5.1.1 Defining Nested Lists, Matrices, and Vectors 330
5.1.2 Extracting Elements of Matrices 335
5.1.3 Basic Computations with Matrices 338
5.1.4 Basic Computations with Vectors 342
5.2 Linear Systems of Equations 350
5.2.1 Calculating Solutions of Linear Systems of Equations 350
5.2.2 Gauss–Jordan Elimination 355
5.3 Selected Topics from Linear Algebra 362
5.3.1 Fundamental Subspaces Associated with Matrices 362
5.3.2 The Gram–Schmidt Process 364
5.3.3 Linear Transformations 368
5.3.4 Eigenvalues and Eigenvectors 371
5.3.5 Jordan Canonical Form 374
5.3.6 The QR Method 377
5.4 Maxima and Minima Using Linear Programming 379
5.4.1 The Standard Form of a Linear Programming Problem 379
5.4.2 The Dual Problem 381
5.5 Selected Topics from Vector Calculus 387
5.5.1 Vector-Valued Functions 387
5.5.2 Line Integrals 397
5.5.3 Surface Integrals 400
5.5.4 A Note on Nonorientability 404
5.5.5 More on Tangents, Normals, and Curvature in R3 417
5.6 Matrices and Graphics 428
5.7 Exercises 443
Chapter 6. Applications Related to Ordinary and Partial Differential Equations 448
6.1 First-Order Differential Equations 448
6.1.1 Separable Equations 448
6.1.2 Linear Equations 455
6.1.3 Nonlinear Equations 463
6.1.4 Numerical Methods 466
6.2 Second-Order Linear Equations 470
6.2.1 Basic Theory 470
6.2.2 Constant Coefficients 471
6.2.3 Undetermined Coefficients 477
6.2.4 Variation of Parameters 483
6.3 Higher-Order Linear Equations 485
6.3.1 Basic Theory 485
6.3.2 Constant Coefficients 486
6.3.3 Undetermined Coefficients 488
6.3.4 Laplace Transform Methods 494
6.3.5 Nonlinear Higher-Order Equations 505
6.4 Systems of Equations 505
6.4.1 Linear Systems 505
6.4.2 Nonhomogeneous Linear Systems 518
6.4.3 Nonlinear Systems 524
6.5 Some Partial Differential Equations 545
6.5.1 The One-Dimensional Wave Equation 545
6.5.2 The Two-Dimensional Wave Equation 550
6.5.3 Other Partial Differential Equations 560
6.6 Exercises 563
References 570
Index 572