A course in mathematical methods for physicists

Herman, Russell L. A course in mathematical methods for physicists (CRC Press,2013)(ISBN 146658467X)(600dpi)(788p) ......Page 3
Copyright ......Page 4
Contents vii ......Page 6
Introduction xiii ......Page 12
What Is Mathematical Physics? xiv ......Page 13
An Overview of the Course xvi ......Page 15
Tips for Students xxiii ......Page 22
Acknowledgments xxiv ......Page 23
1.1.1 Introduction 1 ......Page 25
1.1.2 Trigonometric Functions 3 ......Page 27
1.1.3 Hyperbolic Functions 7 ......Page 31
1.1.4 Derivatives 9 ......Page 33
1.1.5 Integrals 10 ......Page 34
1.1.6 Geometric Series 23 ......Page 47
1.1.7 Power Series 26 ......Page 50
1.1.8 Binomial Expansion 33 ......Page 57
1.2 What I Need from My Intro Physics Class? 39 ......Page 63
1.3 Technology and Tables 41 ......Page 65
1.4 Appendix: Dimensional Analysis 43 ......Page 67
Problems 47 ......Page 71
2.1 Free Fall 53 ......Page 77
2.2 First-Order Differential Equations 56 ......Page 80
2.2.1 Separable Equations 57 ......Page 81
2.2.2 Linear First-Order Equations 58 ......Page 82
2.2.3 Terminal Velocity 60 ......Page 84
2.3.1 Mass-Spring Systems 62 ......Page 86
2.3.2 Simple Pendulum 63 ......Page 87
2.4 Second-Order Linear Differential Equations 64 ......Page 88
2.4.1 Constant Coefficient Equations 65 ......Page 89
2.5 LRC Circuits 68 ......Page 92
2.5.1 Special Cases 69 ......Page 93
2.6 Damped Oscillations 73 ......Page 97
2.7.1 Method of Undetermined Coefficients 75 ......Page 99
2.7.2 Periodically Forced Oscillations 78 ......Page 102
2.7.3 Method of Variation of Parameters 81 ......Page 105
2.7.4 Initial Value Green's Functions 84 ......Page 108
2.8 Cauchy-Euler Equations 89 ......Page 113
2.9 Numerical Solutions of ODEs 93 ......Page 117
2.9.1 Euler's Method 94 ......Page 118
2.9.2 Higher-Order Taylor Methods 98 ......Page 122
2.9.3 Runge-Kutta Methods 102 ......Page 126
2.10.1 Nonlinear Pendulum 107 ......Page 131
2.10.2 Extreme Sky Diving 111 ......Page 135
2.10.3 Flight of Sports Balls 114 ......Page 138
2.10.4 Falling Raindrops 117 ......Page 141
2.10.5 Two-Body Problem 124 ......Page 148
2.10.6 Expanding Universe 130 ......Page 154
2.10.7 Coefficient of Drag 137 ......Page 161
2.11.1 Coupled Oscillators 139 ......Page 163
2.11.2 Planar Systems 141 ......Page 165
2.11.3 Equilibrium Solutions and Nearby Behaviors 143 ......Page 167
2.11.4 Polar Representation of Spirals 151 ......Page 175
Problems 153 ......Page 177
3.1 Finite Dimensional Vector Spaces 161 ......Page 185
3.2 Linear Transformations 167 ......Page 191
3.2.1 Active and Passive Rotations 168 ......Page 192
3.2.2 Rotation Matrices 169 ......Page 193
3.2.3 Matrix Representations 172 ......Page 196
3.2.4 Matrix Inverses and Determinants 177 ......Page 201
3.2.5 Cramer's Rule 181 ......Page 205
3.3.1 An Introduction to Coupled Systems 184 ......Page 208
3.3.2 Eigenvalue Problems 186 ......Page 210
3.3.3 Rotations of Conics 188 ......Page 212
3.4 Matrix Formulation of Planar Systems 193 ......Page 217
3.4.1 Solving Constant Coefficient Systems in 2D 194 ......Page 218
3.4.2 Examples of the Matrix Method 197 ......Page 221
3.5 Applications 202 ......Page 226
3.5.1 Mass-Spring Systems 203 ......Page 227
3.5.2 Circuits 205 ......Page 229
3.5.3 Mixture Problems 206 ......Page 230
3.5.4 Chemical Kinetics * 209 ......Page 233
3.5.6 Love Affairs 210 ......Page 234
3.5.7 Epidemics 211 ......Page 235
3.6 Appendix: Diagonalization and Linear Systems 212 ......Page 236
Problems 215 ......Page 239
4.1 Introduction 221 ......Page 245
4.2 Logistic Equation 222 ......Page 246
4.2.1 Riccati Equation 224 ......Page 248
4.3 Autonomous First-Order Equations 226 ......Page 250
4.4 Bifurcations for First-Order Equations 227 ......Page 251
4.5 Nonlinear Pendulum 230 ......Page 254
4.5.1 Period of the Nonlinear Pendulum 231 ......Page 255
4.6 Stability of Fixed Points in Nonlinear Systems 235 ......Page 259
4.7 Nonlinear Population Models 246 ......Page 270
4.8 Limit Cycles 250 ......Page 274
4.9 Nonautonomous Nonlinear Systems 256 ......Page 280
4.10 Exact Solutions Using Elliptic Functions 261 ......Page 285
Problems 264 ......Page 288
5.1 Harmonics and Vibrations 271 ......Page 311
5.2 Boundary Value Problems 273 ......Page 313
5.3 Partial Differential Equations 274 ......Page 314
5.4 1D Heat Equation 276 ......Page 316
5.5 1D Wave Equation 279 ......Page 319
5.6 Introduction to Fourier Series 282 ......Page 322
5.7 Fourier Trigonometric Series 285 ......Page 325
5.8 Fourier Series over Other Intervals 291 ......Page 331
5.8.1 Fourier Series on [a, b] 297 ......Page 337
5.9 Sine and Cosine Series 299 ......Page 339
5.10 Solution of the Heat Equation 304 ......Page 344
5.11 Finite Length Strings 307 ......Page 347
5.12 Gibbs Phenomenon 308 ......Page 348
5.13 Green's Functions for 1D Partial Differential Equations 313 ......Page 353
5.13.1 Heat Equation 314 ......Page 354
5.13.2 Wave Equation 315 ......Page 355
5.14.1 Derivation of Wave Equation for String 316 ......Page 356
5.14.2 Derivation of 1D Heat Equation 318 ......Page 358
Problems 319 ......Page 359
6 Non-Sinusoidal Harmonics and Special Functions 323 ......Page 363
6.1 Function Spaces 324 ......Page 364
6.2 Classical Orthogonal Polynomials 327 ......Page 367
6.3 Fourier-Legendre Series 330 ......Page 370
6.3.1 Properties of Legendre Polynomials 331 ......Page 371
6.3.2 Generating Functions: Generating Function for Legendre Polynomials 334 ......Page 374
6.3.3 Differential Equation for Legendre Polynomials 338 ......Page 378
6.3.4 Fourier-Legendre Series 339 ......Page 379
6.4 Gamma Function 341 ......Page 381
6.5 Fourier-Bessel Series 343 ......Page 383
6.6 Sturm-Liouville Eigenvalue Problems 347 ......Page 387
6.6.1 Sturm-Liouville Operators 348 ......Page 388
6.6.2 Properties of Sturm-Liouville Eigenvalue Problems 351 ......Page 391
6.6.3 Adjoint Operators 354 ......Page 394
6.6.4 Lagrange's and Green's Identities 356 ......Page 396
6.6.5 Orthogonality and Reality 357 ......Page 397
6.6.6 Rayleigh Quotient 358 ......Page 398
6.6.7 Eigenfunction Expansion Method 360 ......Page 400
6.7.1 Boundary Value Green's Function 362 ......Page 402
6.7.2 Properties of Green's Functions 365 ......Page 405
6.7.3 Differential Equation for the Green's Function 369 ......Page 409
6.7.4 Series Representations of Green's Functions 373 ......Page 413
6.7.5 Nonhomogeneous Heat Equation 378 ......Page 418
6.8 Appendix: Least Squares Approximation 383 ......Page 423
6.9 Appendix: Fredholm Alternative Theorem 385 ......Page 425
Problems 388 ......Page 428
7.1 Complex Representations of Waves 395 ......Page 435
7.2 Complex Numbers 397 ......Page 437
7.3 Complex Valued Functions 400 ......Page 440
7.3.1 Complex Domain Coloring 403 ......Page 443
7.4 Complex Differentiation 407 ......Page 447
7.5.1 Complex Path Integrals 410 ......Page 450
7.5.2 Cauchy's Theorem 413 ......Page 453
7.5.3 Analytic Functions and Cauchy's Integral Formula 417 ......Page 457
7.5.4 Laurent Series 421 ......Page 461
7.5.5 Singularities and Residue Theorem 424 ......Page 464
7.5.6 Infinite Integrals 432 ......Page 472
7.5.7 Integration over Multivalued Functions 438 ......Page 478
7.5.8 Appendix: Jordan's Lemma 442 ......Page 482
Problems 443 ......Page 483
8.1 Introduction 449 ......Page 489
8.1.1 Example 1: Linearized KdV Equation 450 ......Page 490
8.1.2 Example 2: Free Particle Wave Function 452 ......Page 492
8.1.3 Transform Schemes 454 ......Page 494
8.2 Complex Exponential Fourier Series 455 ......Page 495
8.3 Exponential Fourier Transform 457 ......Page 497
8.4 Dirac Delta Function 461 ......Page 501
8.5 Properties of the Fourier Transform 464 ......Page 504
8.5.1 Fourier Transform Examples 466 ......Page 506
8.6 Convolution Operation 471 ......Page 511
8.6.1 Convolution Theorem for Fourier Transforms 474 ......Page 514
8.6.2 Application to Signal Analysis 478 ......Page 518
8.6.3 Parseval's Equality 480 ......Page 520
8.7 Laplace Transform 481 ......Page 521
8.7.1 Properties and Examples of Laplace Transforms 483 ......Page 523
8.8.1 Series Summation Using Laplace Transforms 488 ......Page 528
8.8.2 Solution of ODEs Using Laplace Transforms 491 ......Page 531
8.8.3 SteP and Impulse Functions 494 ......Page 534
8.9 Convolution Theorem 499 ......Page 539
8.10 Inverse Laplace Transform 502 ......Page 542
8.11.1 Fourier Transform and the Heat Equation 505 ......Page 545
8.11.2 Laplace's Equation on the Half Plane 507 ......Page 547
8.11.3 Heat Equation on Infinite Interval, Revisited 509 ......Page 549
8.11.4 Nonhomogeneous Heat Equation 511 ......Page 551
Problems 514 ......Page 554
9 Vector Analysis and EM Waves 521 ......Page 561
9.1.1 A Review of Vector Products 522 ......Page 562
9.1.2 Differentiation and Integration of Vectors 531 ......Page 571
9.1.3 Div, Grad, Curl 535 ......Page 575
9.1.4 Integral Theorems 541 ......Page 581
9.1.5 Vector Identities 546 ......Page 586
9.1.6 Kepler Problem 547 ......Page 587
9.2.1 Maxwell's Equations 553 ......Page 593
9.2.2 Electromagnetic Wave Equation 555 ......Page 595
9.2.3 Potential Functions and Helmholtz's Theorem 557 ......Page 597
9.3 Curvilinear Coordinates 559 ......Page 599
9.4 Tensors 567 ......Page 607
Problems 570 ......Page 610
10.1 Stationary and Extreme Values of Functions 577 ......Page 617
10.1.1 Functions of One Variable 578 ......Page 618
10.1.2 Functions of Several Variables 580 ......Page 620
10.1.3 Linear Regression 586 ......Page 626
10.1.4 Lagrange Multipliers and Constraints 599 ......Page 639
10.2.1 Introduction 605 ......Page 645
10.2.2 Variational Problems 607 ......Page 647
10.2.3 Euler Equation 609 ......Page 649
10.2.4 Isoperimetic Problems 616 ......Page 656
10.3 Hamilton's Principle 623 ......Page 663
10.4 Geodesics 633 ......Page 673
Problems 642 ......Page 682
11 Problems in Higher Dimensions 647 ......Page 687
11.1 Vibrations of Rectangular Membranes 649 ......Page 689
11.2 Vibrations of a Kettle Drum 654 ......Page 694
11.3 Laplace's Equation in 2D 663 ......Page 703
11.3.1 Poisson Integral Formula 671 ......Page 711
11.4 Three-Dimensional Cake Baking 672 ......Page 712
11.5 Laplace's Equation and Spherical Symmetry 679 ......Page 719
11.5.1 Spherical Harmonics 685 ......Page 725
11.6 Schrodinger Equation in Spherical Coordinates 688 ......Page 728
11.7 Solution of the 3D Poisson Equation 694 ......Page 734
11.7.1 Green's Functions for the 2D Poisson Equation 697 ......Page 737
11.8.1 Introduction 700 ......Page 740
11.8.2 Laplace's Equation: V2i/> = 0 704 ......Page 744
11.8.3 Homogeneous Time-Dependent Equations 705 ......Page 745
11.8.4 Inhomogeneous Steady-State Equation 706 ......Page 746
Problems 707 ......Page 747
Appendix Review of Sequences and Infinite Series 711 ......Page 751
A.2 Convergence of Sequences 712 ......Page 752
A.3 Limit Theorems 713 ......Page 753
A.4 Infinite Series 715 ......Page 755
A.5 Convergence Tests 716 ......Page 756
A.6 Sequences of Functions 720 ......Page 760
A.7 Infinite Series of Functions 723 ......Page 763
A.8 Special Series Expansions 725 ......Page 765
A.9 Order of Sequences and Functions 727 ......Page 767
Problems 729 ......Page 769
Bibliography 733 ......Page 773
Index 737 ......Page 777
cover......Page 1