Introduction to Mathematical Methods in Physics

Contents

PREFACE vii

I DISTRIBUTION FUNCTIONS, GRAPHS, AND APPROXIMATIONS 1
1.1. Averages and Deviations 2
1.2. Distribution Functions 7
1.3. Applications of Distribution Functions 11
1.4. Linear Graphs 16
1.5. Least-Squares Fit 20
1.6. Power Series 23
1.7. Applications of Power Series 36
1.8. Complex Numbers and the Euler Identity 43
1.9. Errors and Introduction to Numerical Methods 45
Exercises 55

II FIRST-ORDER DIFFERENTIAL EQUATIONS 64
2.1. First-Order Equations: Separable 65
2.2. First-Order Equations: Exact 81
2.3. First-Order Equations: Linear 91
2.4. Numericallntegration 101
Exercises 113

III SECOND-ORDER DIFFERENTIAL EQUATIONS 120
3.1. Second-Order Equations: Homogeneous 121
3.2. Second-Order Equations: Inhomogeneous 145
3.3. Series Solution of Ordinary Differential Equations 156
3.4. Numerical Solution of Differential Equations 165
3.5. The Laplace Transform Method 184
Exercises 199

IV WAVES AND FOURIER ANALYSIS 206
4.1. Waves 206
4.2. Partial Differentiation 207
4.3. The Wave Equation 213
4.4. Principle of Superposition 220
4.5. Standing Waves and Harmonics 224
4.6. Fourier Series 229
4.7. Parseval\'s Theorem and Frequency Spectra 241
4.8. Solution of Inhomogeneous DEs 246
4.9. Fourier Transforms and the Dirac Delta Function 248
Exercises 257

V VECTORS AND MATRICES 264
5.1. Algebra of Vectors 264
5.2. Basis Vectors and Components 271
5.3. Vector Spaces 275
5.4. Matrix Algebra 278
5.5. Numerical Methods for Matrices 293
5.6. Coordinate Transformations 307
5.7. Four-Vectors 315
5.8. The Eigenvalue Problem 320
Exercises 346

VI VECTOR CALCULUS 356
6.1. Time Derivatives of Vectors 356
6.2. Fluid Kinematics 359
6.3. Fluid Dynamics 361
6.4. Fields and the Gradient 364
6.5. Fluid Flow and the Divergence 368
6.6. Circulation and the Curl 375
6.7. Conservative Forces and the Laplacian 382
6.8. Electric and Magnetic Fields 388
6.9. Vector Calculus Expressions and Identities 397
Exercises 402

VII COMPLEX VARIABLES 409
7.1. Functions of a Complex Variable 409
7.2. Differentiation and Integration 411
7.3. Cauchy Integral Formula and Laurent Expansion 417
7.4. Singularities, Poles, and Residues 423
7.5. Applications 433
Exercises 451

VIII PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 455
8.1. Introduction to Partial Differential Equations 455
8.2. The Wave Equation Revisited 460
8.3. Laplace\'s Equation 471
8.4. Orthogonal Functions and the Sturm-Liouville Problem 482
8.5. The Legendre Equation 488
8.6. The Bessel Equation 499
8.7. The Hermite Equation 522
Exercises 528

APPENDIX A USEFUL INTEGRALS 533
APPENDIX B GAMMA AND ERROR FUNCTIONS 538
APPENDIX C COMPUTER PROGRAMS 542

BIBLIOGRAPHY 558

SOLUTIONS 559

INDEX 562