# Mathematical Physics Book Bruce R. Kusse and Erik A. Westwig  1 A Review of Vector and Matrix Algebra Using
SubscriptlSummationConventions 1
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
Systems, 58

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 TheDirac &Function 100
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), 111
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 135
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

7 Fourier Seriesa 219
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 250
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 339
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

11 Solutions to Laplace’s Equation
11.1 Cartesian Solutions, 424
11.2 Expansions With Eigenfunctions, 433
11.3 Cylindrical Solutions, 441
11.4 Spherical Solutions, 458

12 Integral Equations 491
12.1 Classificationof 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 509
13.1 Multivalued Functions, 509
13.2 The Method of Steepest Descent, 542
14 Tensors in Non-Orthogonal Coordinate Systems 562
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 639
Appendix B The CurvilinearCurl 641
Appendiv C The Double Integral Identity 645
Appendix D Green’s Function Solutions 647

Appendix E Pseudovectors and the Mirror TestAppendix F Christoffel Symbols and Covariant Derivatives

Appendix G Calculus of Variations 661