Introduction to Mathematical Physics

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1 Introduction to Mathematical Physics Methods and Concepts Second Edition Chun Wa Wong Department of Physics and Astronomy University of California Los Angeles OXFORD UNIVERSITY PRESS

2 Contents 1 Vectors and fields in space Concepts of space Vectors in space Permutation symbols Vector differentiation of a scalar field Vector differentiation of a vector field Path-dependent scalar and vector integrations Flux, divergence and Gauss's theorem Circulation, curl and Stokes's theorem Helmholtz's theorem Orthogonal curvilinear coordinate systems Vector differential operators in orthogonal curvilinear coordinate systems 65 Appendix 1 Tables of mathematical formulas 72 2 Transformations, matrices and operators Transformations and the laws of physics Rotations in space: Matrices Determinant and matrix inversion Homogeneous equations The matrix eigenvalue problem Generalized matrix eigenvalue problems Eigenvalues and eigenvectors of Hermitian matrices The wave equation Displacement in time and translation in space: Infinitesimal generators Rotation operators Matrix groups 129 Appendix 2 Tables of mathematical formulas Relativistic square-root spaces* Introduction Special relativity and Lorentz transformations Relativistic kinematics and the mass-energy equivalence Quaternions Dirac equation, spinors and matrices Symmetries of the Dirac equation* 172 'Marks an advanced topic in Contents, or a long or difficult problem in the chapters.

3 X Contents 3.7 Weyl and Majorana spinors, symmetry violations* Lorentz group Cartan spinors and spin transformations in square-root space 3.10 Dyadics 3.11 Cartesian tensors Tensor analysis 217 Appendix 3 Tables of mathematical formulas Fourier series and Fourier transforms Wave-particle duality: Quantum mechanics Fourier series Fourier coefficients and Fourier-series representation Complex Fourier series and the Dirac 5 function Fourier transform Green function and convolution Heisenberg's uncertainty principle Conjugate variables and operators in wave mechanics Generalized Fourier series and Legendre polynomials Orthogonal functions and orthogonal polynomials Mean-square error and mean-square convergence Convergence of Fourier series Maxwell equations in Fourier spaces D Fourier transforms: Helmholtz decomposition theorem 305 Appendix 4A Short table of Fourier cosine series 313 Appendix 4B Short table of Fourier sine series 313 Appendix 4C Short table of Fourier transforms 314 Appendix 4D Short table of 3D and 4D Fourier transforms 314 Appendix 4E Tables of mathematical formulas Differential equations in physics Introduction Linear differential equations First-order differential equations Second-order linear differential equations The second homogeneous solution and an inhomogeneous solution Green functions Series solution of the homogeneous second-order linear differential equation Differential eigenvalue equations and orthogonal functions Partial differential equations of physics Separation of variables and eigenfunction expansions Boundary and initial conditions Separation of variables for the Laplacian Green functions for partial differential equations 364 Appendix 5 Tables of mathematical formulas

4 6 Nonlinear systems* 6.1 Introduction 6.2 Nonlinear instabilities 6.3 Logistic map and chaos 6.4 Strange attractor 6.5 Driven dissipative linear pendula 6.6 Chaos in parametrically driven dissipative nonlinear pendula 6.7 Solitons 6.8 Traveling kinks 6.9 Nonlinear superposition of solitons 6.10 More general methods for multi-solitons* Appendix 6 Tables of mathematical formulas 7 Special functions 7.1 Introduction 7.2 Generating function for Legendre polynomials 7.3 Hermite polynomials and the quantum oscillator 7.4 Orthogonal polynomials 7.5 Classical orthogonal polynomials* 7.6 Associated Legendre polynomials and spherical harmonics 7.7 Bessel functions 7.8 Sturm-Liouville equation and eigenfunction expansions Appendix 7 Tables of mathematical formulas 8 Functions of a complex variable 8.1 Introduction 8.2 Functions of a complex variable 8.3 Multivalued functions and Riemann surfaces 8.4 Complex differentiation: Analytic functions and singularities 8.5 Complex integration: Cauchy integral theorem and integral formula 8.6 Harmonic functions in the plane 8.7 Taylor series and analytic continuation 8.8 Laurent series 8.9 Residues 8.10 Complex integration: Calculus of residues 8.11 Poles on the contour and Green functions 8.12 Laplace transform 8.13 Inverse Laplace transform 8.14 Construction of functions and dispersion relations 8.15 Asymptotic expansions* Appendix 8 Tables of mathematical formulas

5 xii Contents Appendix A Tutorials 620 A.l Complex algebra 620 A.2 Vectors 627 A.3 Simple and partial differentiations 630 A.4 Simple and multiple integrals 636 A.5 Matrices and determinants 643 A.6 Infinite series 650 A.7 Exponential functions 662 Appendix B Mathematica and other computer algebra systems 670 Appendix C Computer algebra (CA) with Mathematica 611 C.l Introduction to CA 677 C.2 Equation solvers 679 C.3 Drawing figures and graphs 683 C.4 Number-intensive calculations 684 Resources for students 688 Bibliography 694 Name index 699 Subject index 702

ADVANCED ENGINEERING MATHEMATICS MATLAB

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