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| Mathematical Physics |
1 Vector and Matrix Algebra
1.1 Notation, I
1.2 Vector Operations, 5
2 Differential and Integral Operations on Vector and Scalar Fields 18
2.1 Plotting Scalar and Vector Fields, 18
2.2 Integral Operators, 20
2.3 Differential Operations, 23
2.4 Integral Definitions of the Differential Operators, 34
2.5 TheTheorems, 35
3 Curvilinear Coordinate Systems
3.1 The Position Vector, 44
3.2 The Cylindrical System, 45
3.3 The Spherical System, 48
3.4 General Curvilinear Systems, 49
3.5 The Gradient, Divergence, and Curl in Cylindrical and Spherical
4 Introduction to Tensors 67
4.1 The Conductivity Tensor and Ohm’s Law, 67
4.2 General Tensor Notation and Terminology, 71
4.3 Transformations Between Coordinate Systems, 7 1
4.4 Tensor Diagonalization, 78
4.5 Tensor Transformations in Curvilinear Coordinate Systems, 84
4.6 Pseudo-Objects, 86
5 The Dirac &Function
5.1 Examples of Singular Functions in Physics, 100
5.2 Two Definitions of &t), 103
5.3 6-Functions with Complicated Arguments, 108
5.4 Integrals and Derivatives of 6(t), 11 1
5.5 Singular Density Functions, 114
5.6 The Infinitesimal Electric Dipole, 121
5.7 Riemann Integration and the Dirac &Function, 125
6 Introduction to Complex Variables
6.1 A Complex Number Refresher, 135
6.2 Functions of a Complex Variable, 138
6.3 Derivatives of Complex Functions, 140
6.4 The Cauchy Integral Theorem, 144
6.5 Contour Deformation, 146
6.6 The Cauchy Integrd Formula, 147
6.7 Taylor and Laurent Series, 150
6.8 The Complex Taylor Series, 153
6.9 The Complex Laurent Series, 159
6.10 The Residue Theorem, 171
6.1 1 Definite Integrals and Closure, 175
6.12 Conformal Mapping, 189
CONTENTS
7 Fourier Series
7.1 The Sine-Cosine Series, 219
7.2 The Exponential Form of Fourier Series, 227
7.3 Convergence of Fourier Series, 231
7.4 The Discrete Fourier Series, 234
8 Fourier Transforms
8.1 Fourier Series as To -+ m, 250
8.2 Orthogonality, 253
8.3 Existence of the Fourier Transform, 254
8.4 The Fourier Transform Circuit, 256
8.5 Properties of the Fourier Transform, 258
8.6 Fourier Transforms-Examples, 267
8.7 The Sampling Theorem, 290
9 Laplace Transforms
9.1 Limits of the Fourier Transform, 303
9.2 The Modified Fourier Transform, 306
9.3 The Laplace Transform, 313
9.4 Laplace Transform Examples, 314
9.5 Properties of the Laplace Transform, 318
9.6 The Laplace Transform Circuit, 327
9.7 Double-Sided or Bilateral Laplace Transforms, 331
10 Differential Equations
10.1 Terminology, 339
10.2 Solutions for First-Order Equations, 342
10.3 Techniques for Second-Order Equations, 347
10.4 The Method of Frobenius, 354
10.5 The Method of Quadrature, 358
10.6 Fourier and Laplace Transform Solutions, 366
10.7 Green’s Function Solutions, 376
X
11 Solutions to Laplace’s Equation
11.1 Cartesian Solutions, 424
1 1.2 Expansions With Eigenfunctions, 433
11.3 Cylindrical Solutions, 441
1 1.4 Spherical Solutions, 458
12 Integral Equations
12.1 Classification of Linear Integral Equations, 492
12.2 The Connection Between Differential and
Integral Equations, 493
12.3 Methods of Solution, 498
13 Advanced Topics in Complex Analysis
13.1 Multivalued Functions, 509
13.2 The Method of Steepest Descent, 542
14 Tensors in Non-Orthogonal Coordinate Systems
14.1 A Brief Review of Tensor Transformations, 562
14.2 Non-Orthononnal Coordinate Systems, 564
15 Introduction to Group Theory
15.1 The Definition of a Group, 597
15.2 Finite Groups and Their Representations, 598
15.3 Subgroups, Cosets, Class, and Character, 607
15.4 Irreducible Matrix Representations, 612
15.5 Continuous Groups, 630
Appendix A The Led-Cidta Identity
Appendix B The Curvilinear Curl
Appendiv C The Double Integral Identity
Appendix D Green’s Function Solutions
Appendix E Pseudovectors and the Mirror Test
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