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```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
1.2.1 Packages Included with Older Versions of
Mathematica .........      ...................    14
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
2.4 Solving Equations ..........    .................. 100
2.4.1 Exact Solutions of Equations .................. 100
2.4.2 Approximate Solutions of Equations ............ 110
2.5  Exercises  ................................          115
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
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 R' . ...  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
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

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Library of Congress subject headings for this publication: Mathematica (Computer file)Mathematics Data processing