Mathematical Physics: Applied Mathematics for Scientists and Engineers

Mathematical Physics: Applied Mathematics for Scientists and Engineers......Page 2
CONTENTS......Page 10
1.1 Notation......Page 16
1.2 Vector Operations......Page 20
2.1 Plotting Scalar and Vector Fields......Page 33
2.2 Integral Operators......Page 35
2.3 Differential Operations......Page 38
2.4 Integral Definitions of the Differential Operators......Page 49
2.5 TheTheorems......Page 50
3.1 The Position Vector......Page 59
3.2 The Cylindrical System......Page 60
3.3 The Spherical System......Page 63
3.4 General Curvilinear Systems......Page 64
3.5 The Gradient, Divergence, and Curl in Cylindrical and Spherical Systems......Page 73
4.1 The Conductivity Tensor and Ohm’s Law......Page 82
4.3 Transformations Between Coordinate Systems......Page 86
4.4 Tensor Diagonalization......Page 93
4.5 Tensor Transformations in Curvilinear Coordinate Systems......Page 99
4.6 Pseudo-Objects......Page 101
5.1 Examples of Singular Functions in Physics......Page 115
5.2 Two Definitions of δ(t)......Page 118
5.3 6 δ-Functions with Complicated Arguments......Page 123
5.4 Integrals and Derivatives of δ(t)......Page 126
5.5 Singular Density Functions......Page 129
5.6 The Infinitesimal Electric Dipole......Page 136
5.7 Riemann Integration and the Dirac δ-Function......Page 140
6.1 A Complex Number Refresher......Page 150
6.2 Functions of a Complex Variable......Page 153
6.3 Derivatives of Complex Functions......Page 155
6.4 The Cauchy Integral Theorem......Page 159
6.5 Contour Deformation......Page 161
6.6 The Cauchy Integral Formula......Page 162
6.7 Taylor and Laurent Series......Page 165
6.8 The Complex Taylor Series......Page 168
6.9 The Complex Laurent Series......Page 174
6.10 The Residue Theorem......Page 186
6.11 Definite Integrals and Closure......Page 190
6.12 Conformal Mapping......Page 204
7.1 The Sine-Cosine Series......Page 234
7.2 The Exponential Form of Fourier Series......Page 242
7.3 Convergence of Fourier Series......Page 246
7.4 The Discrete Fourier Series......Page 249
8.1 Fourier Series as T0 → ∞......Page 265
8.2 Orthogonality......Page 268
8.3 Existence of the Fourier Transform......Page 269
8.4 The Fourier Transform Circuit......Page 271
8.5 Properties of the Fourier Transform......Page 273
8.6 Fourier Transforms-Examples......Page 282
8.7 The Sampling Theorem......Page 305
9.1 Limits of the Fourier Transform......Page 318
9.2 The Modified Fourier Transform......Page 321
9.3 The Laplace Transform......Page 328
9.4 Laplace Transform Examples......Page 329
9.5 Properties of the Laplace Transform......Page 333
9.6 The Laplace Transform Circuit......Page 342
9.7 Double-Sided or Bilateral Laplace Transforms......Page 346
10.1 Terminology......Page 354
10.2 Solutions for First-Order Equations......Page 357
10.3 Techniques for Second-Order Equations......Page 362
10.4 The Method of Frobenius......Page 369
10.5 The Method of Quadrature......Page 373
10.6 Fourier and Laplace Transform Solutions......Page 381
10.7 Green’s Function Solutions......Page 391
11.1 Cartesian Solutions......Page 439
11.2 Expansions With Eigenfunctions......Page 448
11.3 Cylindrical Solutions......Page 456
11.4 Spherical Solutions......Page 473
12 Integral Equations......Page 506
12.1 Classification of Linear Integral Equations......Page 507
12.2 The Connection Between Differential and Integral Equations......Page 508
12.3 Methods of Solution......Page 513
13.1 Multivalued Functions......Page 524
13.2 The Method of Steepest Descent......Page 557
14.1 A Brief Review of Tensor Transformations......Page 577
14.2 Non-Orthononnal Coordinate Systems......Page 579
15.1 The Definition of a Group......Page 612
15.2 Finite Groups and Their Representations......Page 613
15.3 Subgroups, Cosets, Class, and Character......Page 622
15.4 Irreducible Matrix Representations......Page 627
15.5 Continuous Groups......Page 645
Appendix A The Levi-Civita Identity......Page 654
Appendix B The Curvilinear Curl......Page 656
Appendiv C The Double Integral Identity......Page 660
Appendix D Green’s Function Solutions......Page 662
Appendix E Pseudovectors and the Mirror Test......Page 668
Appendix F Christoffel Symbols and Covariant Derivatives......Page 670
Appendix G Calculus of Variations......Page 676
Errata List......Page 680
Bibliography......Page 686
Index......Page 688