Note: Cover -- Contents -- 1 Vectors and fields in space -- 1.1 Concepts of space -- 1.2 Vectors in space -- 1.3 Permutation symbols -- 1.4 Vector differentiation of a scalar field -- 1.5 Vector differentiation of a vector field -- 1.6 Path-dependent scalar and vector integrations -- 1.7 Flux, divergence and Gauss's theorem -- 1.8 Circulation, curl and Stokes's theorem -- 1.9 Helmholtz's theorem -- 1.10 Orthogonal curvilinear coordinate systems -- 1.11 Vector differential operators in orthogonal curvilinear coordinate systems -- Appendix 1 Tables of mathematical formulas -- 2 Transformations, matrices and operators -- 2.1 Transformations and the laws of physics -- 2.2 Rotations in space: Matrices -- 2.3 Determinant and matrix inversion -- 2.4 Homogeneous equations -- 2.5 The matrix eigenvalue problem -- 2.6 Generalized matrix eigenvalue problems -- 2.7 Eigenvalues and eigenvectors of Hermitian matrices -- 2.8 The wave equation -- 2.9 Displacement in time and translation in space: Infinitesimal generators -- 2.10 Rotation operators -- 2.11 Matrix groups -- Appendix 2 Tables of mathematical formulas -- 3 Relativistic square-root spaces[sup(*)] -- 3.1 Introduction -- 3.2 Special relativity and Lorentz transformations -- 3.3 Relativistic kinematics and the mass-energy equivalence -- 3.4 Quaternions -- 3.5 Dirac equation, spinors and matrices -- 3.6 Symmetries of the Dirac equation[sup(*)] -- 3.7 Weyl and Majorana spinors, symmetry violations[sup(*)] -- 3.8 Lorentz group -- 3.9 Cartan spinors and spin transformations in square-root space -- 3.10 Dyadics -- 3.11 Cartesian tensors -- 3.12 Tensor analysis -- Appendix 3 Tables of mathematical formulas -- 4 Fourier series and Fourier transforms -- 4.1 Wave-particle duality: Quantum mechanics -- 4.2 Fourier series -- 4.3 Fourier coeffcients and Fourier-series representation. , 4.4 Complex Fourier series and the Dirac & -- #916 -- function -- 4.5 Fourier transform -- 4.6 Green function and convolution -- 4.7 Heisenberg's uncertainty principle -- 4.8 Conjugate variables and operators in wave mechanics -- 4.9 Generalized Fourier series and Legendre polynomials -- 4.10 Orthogonal functions and orthogonal polynomials -- 4.11 Mean-square error and mean-square convergence -- 4.12 Convergence of Fourier series -- 4.13 Maxwell equations in Fourier spaces -- 4.14 3D Fourier transforms: Helmholtz decomposition theorem -- Appendix 4A: Short table of Fourier cosine series -- Appendix 4B: Short table of Fourier sine series -- Appendix 4C: Short table of Fourier transforms -- Appendix 4D: Short table of 3D and 4D Fourier transforms -- Appendix 4E: Tables of mathematical formulas -- 5 Differential equations in physics -- 5.1 Introduction -- 5.2 Linear differential equations -- 5.3 First-order differential equations -- 5.4 Second-order linear differential equations -- 5.5 The second homogeneous solution and an inhomogeneous solution -- 5.6 Green functions -- 5.7 Series solution of the homogeneous second-order linear differential equation -- 5.8 Differential eigenvalue equations and orthogonal functions -- 5.9 Partial differential equations of physics -- 5.10 Separation of variables and eigenfunction expansions -- 5.11 Boundary and initial conditions -- 5.12 Separation of variables for the Laplacian -- 5.13 Green functions for partial differential equations -- Appendix 5 Tables of mathematical formulas -- 6 Nonlinear systems[sup(*)] -- 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[sup(*)] -- 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[sup(*)] -- 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[sup(*)] -- Appendix 8 Tables of mathematical formulas -- Appendix A: Tutorials -- A.1 Complex algebra -- A.2 Vectors -- A.3 Simple and partial differentiations -- A.4 Simple and multiple integrals -- A.5 Matrices and determinants -- A.6 Infinite series -- A.7 Exponential functions -- Appendix B: Mathematica and other computer algebra systems -- Appendix C: Computer algebra (CA) with Mathematica -- C.1 Introduction to CA -- C.2 Equation solvers -- C.3 Drawing figures and graphs -- C.4 Number-intensive calculations -- Resources for students -- Bibliography -- Name index -- A -- B -- C -- D -- E -- F -- G -- H -- J -- K -- L -- M -- N -- P -- R -- S -- T -- V -- W -- Y -- Z -- Subject index -- A. , B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W.