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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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