# Mathematical Methods in Engineering and Physics

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# Mathematical Methods in Engineering and Physics

ISBN: 978-1-118-44960-8 April 2015 830 Pages

## Description

This text is intended for the undergraduate course in math methods, with an audience of physics and engineering majors. As a required course in most departments, the text relies heavily on explained examples, real-world applications and student engagement. Supporting the use of active learning, a strong focus is placed upon physical motivation combined with a versatile coverage of topics that can be used as a reference after students complete the course.

Each chapter begins with an overview that includes a list of prerequisite knowledge, a list of skills that will be covered in the chapter, and an outline of the sections. Next comes the motivating exercise, which steps the students through a real-world physical problem that requires the techniques taught in each chapter.

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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)

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)

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)

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

4.7 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)

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)

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

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)

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

8.6 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

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)

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

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

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)

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)

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

"[Mathematical Methods in Engineering and Physics] is my book of choice for teaching undergraduates...I honestly never thought that I could be so enchanted by the heat equation before seeing how Felder and Felder effectively have students derive it as part of honing their intuition for how to think about partial differential equations." - Christine Aidala, PhD, Associate Professor of Physics at University of Michigan for the American Journal of Physics

Breadth and Depth of Problems. This book has numerous problems – over 2700. Some of them are the sort of rote drill-and-practice thing that everyone provides. Many are aimed at physical application. Many make students probe deeper into the mathematics of what is being presented.

Math is Always Presented in a Physical Context. This text always presents topics as "here is a real problem that someone might actually want to solve, and here is how this mathematical tool helps."  Mathematicians write math methods texts that start with a theorem, then prove it, and provide a context for it. This text, by contrast, always begins with a physical problem. The authors present tools for solving the problem and show you how to use those tools. Then, as the last step—often in a problem for the student, but sometimes in an explanation—they provide the proof.

Exercises/Active Learning. Exercises are very different from problem sets. They are meant to step a student through a process, at the end of which the student has figured out a key mathematical idea on his/her own. "Active learning"—getting students engaged instead of just lecturing at them—is difficult and time-consuming. This text is designed to help with that process.

Computers. You can use this text without ever mentioning computers—the Explanations don't require them, and the Exercises and Problems that do require computers are clearly marked so they can be skipped. But for people who do want to use computers, this text offers several special problems.

Independence of Topics. Whenever possible, the authors avoided making one chapter depend upon another. Addressing how professors may not be able to teach the entire book, the authors provide a freedom to pick and choose the chapters they want to teach.