Mathematica by Example

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