Gary N. Felder and Kenny M. Felder are the authors of Mathematical Methods in Engineering and Physics, published by Wiley.

PREFACE xi

1 Introduction to Ordinary Differential Equations 1

1.1 Motivating Exercise: The Simple Harmonic Oscillator 2

1.2 Overview of Differential Equations 3

1.3 Arbitrary Constants 15

1.4 Slope Fields and Equilibrium 25

1.5 Separation of Variables 34

1.6 Guess and Check, and Linear Superposition 39

1.7 Coupled Equations (see felderbooks.com)

1.8 Differential Equations on a Computer (see felderbooks.com)

1.9 Additional Problems (see felderbooks.com)

2 Taylor Series and Series Convergence 50

2.1 Motivating Exercise: Vibrations in a Crystal 51

2.2 Linear Approximations 52

2.3 Maclaurin Series 60

2.4 Taylor Series 70

2.5 Finding One Taylor Series from Another 76

2.6 Sequences and Series 80

2.7 Tests for Series Convergence 92

2.8 Asymptotic Expansions (see felderbooks.com)

2.9 Additional Problems (see felderbooks.com)

3 Complex Numbers 104

3.1 Motivating Exercise: The Underdamped Harmonic Oscillator 104

3.2 Complex Numbers 105

3.3 The Complex Plane 113

3.4 Euler’s Formula I—The Complex Exponential Function 117

3.5 Euler’s Formula II—Modeling Oscillations 126

3.6 Special Application: Electric Circuits (see felderbooks.com)

3.7 Additional Problems (see felderbooks.com)

4 Partial Derivatives 136

4.1 Motivating Exercise: The Wave Equation 136

4.2 Partial Derivatives 137

4.3 The Chain Rule 145

4.4 Implicit Differentiation 153

4.5 Directional Derivatives 158

4.6 The Gradient 163

4.7 Tangent Plane Approximations and Power Series (see felderbooks.com)

4.8 Optimization and the Gradient 172

4.9 Lagrange Multipliers 181

4.10 Special Application: Thermodynamics (see felderbooks.com)

4.11 Additional Problems (see felderbooks.com)

5 Integrals in Two or More Dimensions 188

5.1 Motivating Exercise: Newton’s Problem (or) The Gravitational Field of a Sphere 188

5.2 Setting Up Integrals 189

5.3 Cartesian Double Integrals over a Rectangular Region 204

5.4 Cartesian Double Integrals over a Non-Rectangular Region 211

5.5 Triple Integrals in Cartesian Coordinates 216

5.6 Double Integrals in Polar Coordinates 221

5.7 Cylindrical and Spherical Coordinates 229

5.8 Line Integrals 240

5.9 Parametrically Expressed Surfaces 249

5.10 Surface Integrals 253

5.11 Special Application: Gravitational Forces (see felderbooks.com)

5.12 Additional Problems (see felderbooks.com)

6 Linear Algebra I 266

6.1 The Motivating Example on which We’re Going to Base the Whole Chapter: The Three-Spring Problem 266

6.2 Matrices: The Easy Stuff 276

6.3 Matrix Times Column 280

6.4 Basis Vectors 286

6.5 Matrix Times Matrix 294

6.6 The Identity and Inverse Matrices 303

6.7 Linear Dependence and the Determinant 312

6.8 Eigenvectors and Eigenvalues 325

6.9 Putting It Together: Revisiting the Three-Spring Problem 336

6.10 Additional Problems (see felderbooks.com)

7 Linear Algebra II 346

7.1 Geometric Transformations 347

7.2 Tensors 358

7.3 Vector Spaces and Complex Vectors 369

7.4 Row Reduction (see felderbooks.com)

7.5 Linear Programming and the Simplex Method (see felderbooks.com)

7.6 Additional Problems (see felderbooks.com)

8 Vector Calculus 378

8.1 Motivating Exercise: Flowing Fluids 378

8.2 Scalar and Vector Fields 379

8.3 Potential in One Dimension 387

8.4 From Potential to Gradient 396

8.5 From Gradient to Potential: The Gradient Theorem 402

8.6 Divergence, Curl, and Laplacian 407

8.7 Divergence and Curl II—The Math Behind the Pictures 416

8.8 Vectors in Curvilinear Coordinates 419

8.9 The Divergence Theorem 426

8.10 Stokes’ Theorem 432

8.11 Conservative Vector Fields 437

8.12 Additional Problems (see felderbooks.com)

9 Fourier Series and Transforms 445

9.1 Motivating Exercise: Discovering Extrasolar Planets 445

9.2 Introduction to Fourier Series 447

9.3 Deriving the Formula for a Fourier Series 457

9.4 Different Periods and Finite Domains 459

9.5 Fourier Series with Complex Exponentials 467

9.6 Fourier Transforms 472

9.7 Discrete Fourier Transforms (see felderbooks.com)

9.8 Multivariate Fourier Series (see felderbooks.com)

9.9 Additional Problems (see felderbooks.com)

10 Methods of Solving Ordinary Differential Equations 484

10.1 Motivating Exercise: A Damped, Driven Oscillator 485

10.2 Guess and Check 485

10.3 Phase Portraits (see felderbooks.com)

10.4 Linear First-Order Differential Equations (see felderbooks.com)

10.5 Exact Differential Equations (see felderbooks.com)

10.6 Linearly Independent Solutions and the Wronskian (see felderbooks.com)

10.7 Variable Substitution 494

10.8 Three Special Cases of Variable Substitution 505

10.9 Reduction of Order and Variation of Parameters (see felderbooks.com)

10.10 Heaviside, Dirac, and Laplace 512

10.11 Using Laplace Transforms to Solve Differential Equations 522

10.12 Green’s Functions 531

10.13 Additional Problems (see felderbooks.com)

11 Partial Differential Equations 541

11.1 Motivating Exercise: The Heat Equation 542

11.2 Overview of Partial Differential Equations 544

11.3 Normal Modes 555

11.4 Separation of Variables—The Basic Method 567

11.5 Separation of Variables—More than Two Variables 580

11.6 Separation of Variables—Polar Coordinates and Bessel Functions 589

11.7 Separation of Variables—Spherical Coordinates and Legendre Polynomials 607

11.8 Inhomogeneous Boundary Conditions 616

11.9 The Method of Eigenfunction Expansion 623

11.10 The Method of Fourier Transforms 636

11.11 The Method of Laplace Transforms 646

11.12 Additional Problems (see felderbooks.com)

12 Special Functions and ODE Series Solutions 652

12.1 Motivating Exercise: The Circular Drum 652

12.2 Some Handy Summation Tricks 654

12.3 A Few Special Functions 658

12.4 Solving Differential Equations with Power Series 666

12.5 Legendre Polynomials 673

12.6 The Method of Frobenius 682

12.7 Bessel Functions 688

12.8 Sturm-Liouville Theory and Series Expansions 697

12.9 Proof of the Orthgonality of Sturm-Liouville Eigenfunctions (see felderbooks.com)

12.10 Special Application: The Quantum Harmonic Oscillator and Ladder Operators (see felderbooks.com)

12.11 Additional Problems (see felderbooks.com)

13 Calculus with Complex Numbers 708

13.1 Motivating Exercise: Laplace’s Equation 709

13.2 Functions of Complex Numbers 710

13.3 Derivatives, Analytic Functions, and Laplace’s Equation 716

13.4 Contour Integration 726

13.5 Some Uses of Contour Integration 733

13.6 Integrating Along Branch Cuts and Through Poles (see felderbooks.com)

13.7 Complex Power Series 742

13.8 Mapping Curves and Regions 747

13.9 Conformal Mapping and Laplace’s Equation 754

13.10 Special Application: Fluid Flow (see felderbooks.com)

13.11 Additional Problems (see felderbooks.com)

Appendix A Different Types of Differential Equations 765

Appendix B Taylor Series 768

AppendixC Summary of Tests for Series Convergence 770

Appendix D Curvilinear Coordinates 772

Appendix E Matrices 774

Appendix F Vector Calculus 777

AppendixG Fourier Series and Transforms 779

Appendix H Laplace Transforms 782

Appendix I Summary: Which PDE Technique Do I Use? 787

Appendix J Some Common Differential Equations and Their Solutions 790

Appendix K Special Functions 798

Appendix L Answers to “Check Yourself” in Exercises 801

AppendixM Answers to Odd-Numbered Problems (see felderbooks.com)

Index 805

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