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Full 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.
Contents
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
Appendix C Summary of Tests for Series Convergence 770
Appendix D Curvilinear Coordinates 772
Appendix E Matrices 774
Appendix F Vector Calculus 777
Appendix G 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
Appendix M Answers to Odd-Numbered Problems (see felderbooks.com)
Index 805
