Mathematical Methods in Physics, Engineering, and Chemistry

BRETT BORDEN, PHD, joined the Faculty of the Naval Postgraduate School in Monterey, CA in 2002, where he is Emeritus Professor of Physics. He served on the editorial board of the journal Inverse Problems from 2002 to 2013. Dr. Borden is a Fellow of The Institute of Physics. JAMES LUSCOMBE, PHD, is a Professor at the Naval Postgraduate School in Monterey, California. He teaches a variety of topics, including general relativity, statistical mechanics, mathematical methods, and quantum computation.

• Preface xi1 Vectors and linear operators 11.1 The linearity of physical phenomena 11.2 Vector spaces 21.2.1 A word on notation 41.2.2 Linear independence, bases, and dimensionality 51.2.3 Subspaces 71.2.4 Isomorphism of N-dimensional spaces 81.2.5 Dual spaces 81.3 Inner products and orthogonality 101.3.1 Inner products 101.3.2 The Schwarz inequality 111.3.3 Vector norms 121.3.4 Orthonormal bases and the Gram–Schmidt process 121.3.5 Complete sets of orthonormal vectors 151.4 Operators and matrices 161.4.1 Linear operators 171.4.2 Representing operators with matrices 181.4.3 Matrix algebra 201.4.4 Rank and nullity 221.4.5 Bounded operators 231.4.6 Inverses 241.4.7 Change of basis and the similarity transformation 251.4.8 Adjoints and Hermitian operators 271.4.9 Determinants and the matrix inverse 291.4.10 Unitary operators 331.4.11 The trace of a matrix 351.5 Eigenvectors and their role in representing operators 361.5.1 Eigenvectors and eigenvalues 361.5.2 The eigenproblem for Hermitian and unitary operators 391.5.3 Diagonalizing matrices 401.6 Hilbert space: Infinite-dimensional vector space 43Exercises 472 Sturm–Liouville theory 512.1 Second-order differential equations 522.1.1 Uniqueness and linear independence 522.1.2 The adjoint operator 552.1.3 Self-adjoint operator 562.2 Sturm–Liouville systems 572.3 The Sturm–Liouville eigenproblem 602.4 The Dirac delta function 642.5 Completeness 662.6 Recap 68Summary 68Exercises 693 Partial differential equations 713.1 A survey of partial differential equations 713.1.1 The continuity equation 713.1.2 The diffusion equation 723.1.3 The free-particle Schrödinger equation 733.1.4 The heat equation 733.1.5 The inhomogeneous diffusion equation 743.1.6 Schrödinger equation for a particle in a potential field 743.1.7 The Poisson equation 743.1.8 The Laplace equation 753.1.9 The wave equation 753.1.10 Inhomogeneous wave equation 763.1.11 Summary of PDEs 763.2 Separation of variables and the Helmholtz equation 763.2.1 Rectangular coordinates 783.2.2 Cylindrical coordinates 803.2.3 Spherical coordinates 823.3 The paraxial approximation 833.4 The three types of linear PDEs 843.4.1 Hyperbolic PDEs 853.4.2 Parabolic PDEs 873.4.3 Elliptic PDEs 873.5 Outlook 88Summary 88Exercises 894 Fourier analysis 914.1 Fourier series 914.2 The exponential form of Fourier series 964.3 General intervals 984.4 Parseval’s theorem 1034.5 Back to the delta function 1054.6 Fourier transform 1074.7 Convolution integral 111Summary 115Exercises 1165 Series solutions of ordinary differential equations 1215.1 The Frobenius method 1225.1.1 Power series 1225.1.2 Introductory example 1235.1.3 Ordinary points 1255.1.4 Regular singular points 1305.2 Wronskian method for obtaining a second solution 1375.3 Bessel and Neumann functions 1375.4 Legendre polynomials 142Summary 144Exercises 1456 Spherical harmonics 1476.1 Properties of the Legendre polynomials, Pl(x) 1486.1.1 Rodrigues formula 1486.1.2 Orthogonality 1506.1.3 Completeness 1516.1.4 Generating function 1526.1.5 Recursion relations 1556.2 Associated Legendre functions, Pm l (x) 1576.3 Spherical harmonic functions, Yml (θ, φ) 1586.4 Addition theorem for Ym l (θ, φ) 1606.5 Laplace equation in spherical coordinates 166Summary 167Exercises 1687 Bessel functions 1737.1 Small-argument and asymptotic forms 1737.1.1 Limiting forms for small argument 1737.1.2 Asymptotic forms for large argument 1747.1.3 Hankel functions 1747.2 Properties of the Bessel functions, Jn(x) 1757.2.1 Series associated with the generating function 1757.2.2 Recursion relations 1777.2.3 Integral representation 1787.3 Orthogonality 1807.4 Bessel series 1827.5 The Fourier-Bessel transform 1857.6 Spherical Bessel functions 1867.6.1 Reduction to elementary functions 1867.6.2 Small-argument forms 1887.6.3 Asymptotic forms 1887.6.4 Orthogonality and completeness 1897.7 Expansion of plane waves in spherical harmonics 190Summary 192Exercises 1928 Complex analysis 1958.1 Complex functions 1958.2 Analytic functions: differentiable in a region 1978.2.1 Continuity, differentiability, and analyticity 1978.2.2 Cauchy–Riemann conditions 1988.2.3 Analytic functions are functions only of z = x + iy 2018.2.4 Useful definitions 2018.3 Contour integrals 2028.4 Integrating analytic functions 2068.5 Cauchy integral formulas 2108.5.1 Derivatives of analytic functions 2118.5.2 Consequences of the Cauchy formulas 2128.6 Taylor and Laurent series 2138.6.1 Taylor series 2138.6.2 The zeros of analytic functions are isolated 2158.6.3 Laurent series 2158.7 Singularities and residues 2178.7.1 Isolated singularities, residue theorem 2178.7.2 Multivalued functions, branch points, and branch cuts 2208.8 Definite integrals 2218.8.1 Integrands containing cos θ and sin θ 2228.8.2 Infinite integrals 2238.8.3 Poles on the contour of integration 2268.9 Meromorphic functions 2288.10 Approximation of integrals 2308.10.1 The method of steepest descent 2338.10.2 The method of stationary phase 2358.11 The analytic signal 2368.11.1 The Hilbert transform 2378.11.2 Paley–Wiener and Titchmarsh theorems 2398.11.3 Is the analytic signal, analytic? 2418.12 The Laplace transform 242Summary 245Exercises 2459 Inhomogeneous differential equations 2519.1 The method of Green functions 2519.1.1 Boundary conditions 2529.1.2 Reciprocity relation: G(x, x’) = G(x’, x) 2539.1.3 Matching conditions 2549.1.4 Direct construction of G(x, x’) 2559.1.5 Eigenfunction expansions 2579.2 Poisson equation 2609.2.1 Boundary conditions and reciprocity relations 2619.2.2 So, what’s the Green function? 2639.3 Helmholtz equation 2669.3.1 Green function for two-dimensional problems 2679.3.2 Free-space Green function for three dimensions 2709.3.3 Expansion in spherical harmonics 2709.4 Diffusion equation 2729.4.1 Boundary conditions, causality, and reciprocity 2729.4.2 Solution to the diffusion equation 2749.4.3 Free-space Green function 2759.5 Wave equation 2799.6 The Kirchhoff integral theorem 283Summary 284Exercises 28410 Integral equations 28710.1 Introduction 28710.1.1 Equivalence of integral and differential equations 28710.1.2 Role of coordinate systems in capturing boundary data 28810.2 Classification of integral equations 29010.3 Neumann series 29110.4 Integral transform methods 29310.4.1 Difference kernels 29310.4.2 Fourier kernels 29410.5 Separable kernels 29510.6 Self-adjoint kernels 29710.7 Numerical approaches 30210.7.1 Matrix form 30210.7.2 Measurement space 30310.7.3 The generalized inverse 306Summary 314Exercises 31511 Tensor analysis 31911.1 Once over lightly: A quick intro to tensors 31911.2 Transformation properties 32711.2.1 The two types of vector: Contravariant and covariant 32711.2.2 Coordinate transformations 32811.2.3 Contravariant vectors and tensors 33211.2.4 Covariant vectors and tensors 33611.2.5 Mixed tensors 33911.2.6 Covariant equations 33911.3 Contraction and the quotient theorem 34011.4 The metric tensor 34211.5 Raising and lowering indices 34411.6 Geometric properties of covariant vectors 34711.7 Relative tensors 35011.8 Tensors as operators 35311.9 Symmetric and antisymmetric tensors 35611.10 The Levi-Civita tensor 35711.11 Pseudotensors 36011.12 Covariant differentiation of tensors 363Summary 373Exercises 374A Vector calculus 377A.1 Scalar fields 377A.1.1 The directional derivative 377A.1.2 The gradient 378A.2 Vector fields 379A.2.1 Divergence 379A.2.2 Curl 380A.2.3 The Laplacian 380A.2.4 Vector operator formulae 381A.3 Integration 382A.3.1 Line integrals 382A.3.2 Surface integrals 383A.4 Important integral theorems in vector calculus 384A.4.1 Green’s theorem in the plane 384A.4.2 The divergence theorem 386A.4.3 Stokes’ theorem 386A.4.4 Conservative fields 387A.4.5 The Helmholtz theorem 389A.5 Coordinate systems 390A.5.1 Orthogonal curvilinear coordinates 390A.5.2 Unit vectors 391A.5.3 Differential displacement 392A.5.4 Differential surface and volume elements 393A.5.5 Transformation of vector components 393A.5.6 Cylindrical coordinates 394B Power series 401C The gamma function, Γ(x) 403Recursion relation 403Limit formula 404Reflection formula 405Digamma function 405D Boundary conditions for Partial Differential Equations 409Summary 417References 419Index 421