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
Contents 1 Vectors and fields in space 1 1.1 Concepts of space 1 1.2 Vectors in space 4 1.3 Permutation symbols 14 1.4 Vector differentiation of a scalar field 20 1.5 Vector differentiation of a vector field 25 1.6 Path-dependent scalar and vector integrations 31 1.7 Flux, divergence and Gauss's theorem 42 1.8 Circulation, curl and Stokes's theorem 48 1.9 Helmholtz's theorem 53 1.10 Orthogonal curvilinear coordinate systems 56 1.11 Vector differential operators in orthogonal curvilinear coordinate systems 65 Appendix 1 Tables of mathematical formulas 72 2 Transformations, matrices and operators 76 2.1 Transformations and the laws of physics 76 2.2 Rotations in space: Matrices 77 2.3 Determinant and matrix inversion 87 2.4 Homogeneous equations 93 2.5 The matrix eigenvalue problem 97 2.6 Generalized matrix eigenvalue problems 104 2.7 Eigenvalues and eigenvectors of Hermitian matrices 108 2.8 The wave equation 114 2.9 Displacement in time and translation in space: Infinitesimal generators 117 2.10 Rotation operators 125 2.11 Matrix groups 129 Appendix 2 Tables of mathematical formulas 135 3 Relativistic square-root spaces* 138 3.1 Introduction 138 3.2 Special relativity and Lorentz transformations 139 3.3 Relativistic kinematics and the mass-energy equivalence 150 3.4 Quaternions 159 3.5 Dirac equation, spinors and matrices 165 3.6 Symmetries of the Dirac equation* 172 'Marks an advanced topic in Contents, or a long or difficult problem in the chapters.
X Contents 3.7 Weyl and Majorana spinors, symmetry violations* 179 3.8 Lorentz group 188 3.9 Cartan spinors and spin transformations in square-root space 3.10 Dyadics 3.11 Cartesian tensors 206 3.12 Tensor analysis 217 Appendix 3 Tables of mathematical formulas 232 4 Fourier series and Fourier transforms 244 4.1 Wave-particle duality: Quantum mechanics 244 4.2 Fourier series 247 4.3 Fourier coefficients and Fourier-series representation 250 4.4 Complex Fourier series and the Dirac 5 function 258 4.5 Fourier transform 265 4.6 Green function and convolution 269 4.7 Heisenberg's uncertainty principle 273 4.8 Conjugate variables and operators in wave mechanics 276 4.9 Generalized Fourier series and Legendre polynomials 280 4.10 Orthogonal functions and orthogonal polynomials 287 4.11 Mean-square error and mean-square convergence 292 4.12 Convergence of Fourier series 295 4.13 Maxwell equations in Fourier spaces 299 4.14 3D 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 315 5 Differential equations in physics 319 5.1 Introduction 319 5.2 Linear differential equations 321 5.3 First-order differential equations 324 5.4 Second-order linear differential equations 328 5.5 The second homogeneous solution and an inhomogeneous solution 332 5.6 Green functions 337 5.7 Series solution of the homogeneous second-order linear differential equation 342 5.8 Differential eigenvalue equations and orthogonal functions 347 5.9 Partial differential equations of physics 350 5.10 Separation of variables and eigenfunction expansions 351 5.11 Boundary and initial conditions 354 5.12 Separation of variables for the Laplacian 359 5.13 Green functions for partial differential equations 364 Appendix 5 Tables of mathematical formulas 368 195 200
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
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