Mathematical Methods for Physicists: A Concise Introduction

by Tai L. Chow

• ISBN13: 9780521652278
• ISBN10: 0521652278
• Format: Hardcover
• Copyright: 2000-07-31
• Publisher: Cambridge University Press

Summary

This text is designed for an intermediate-level, two-semester undergraduate course in mathematical physics. It provides an accessible account of most of the current, important mathematical tools required in physics. The book bridges the gap between an introductory physics course and more advanced courses in classical mechanics, electricity and magnetism, quantum mechanics, and thermal and statistical physics. It contains a large number of worked examples to illustrate the mathematical techniques developed and to show their relevance to physics. The highly organized coverage allows instructors to teach the basics in one semester. The book could also be used in courses in engineering, astronomy, and mathematics.

Table of Contents

Preface	xv
Vector and tensor analysis	1	(61)
Vectors and scalars	1	(2)
Direction angles and direction cosines	3	(1)
Vector algebra	4	(1)
Equality of vectors	4	(1)
Vector addition	4	(1)
Multiplication by a scalar	4	(1)
The scalar product	5	(2)
The vector (cross or outer) product	7	(3)
The triple scalar product A . (B x C)	10	(1)
The triple vector product	11	(1)
Change of coordinate system	11	(2)
The linear vector space Vn	13	(2)
Vector differentiation	15	(1)
Space curves	16	(1)
Motion in a plane	17	(1)
A vector treatment of classical orbit theory	18	(2)
Vector differential of a scalar field and the gradient	20	(1)
Conservative vector field	21	(1)
The vector differential operator ▾	22	(1)
Vector differentiation of a vector field	22	(5)
The divergence of a vector	22	(2)
The operator ▾2, the Laplacian	24	(1)
The curl of a vector	24	(3)
Formulas involving ▾	27	(1)
Orthogonal curvilinear coordinates	27	(5)
Special orthogonal coordinate systems	32	(3)
Cylindrical coordinates (ρ, &phis;, z)	32	(2)
Spherical coordinates (r, &thetas;, &phis;)	34	(1)
Vector integration and integral theorems	35	(9)
Gauss' theorem (the divergence theorem)	37	(2)
Continuity equation	39	(1)
Stokes' theorem	40	(3)
Green's theorem	43	(1)
Green's theorem in the plane	44	(1)
Helmholtz's theorem	44	(1)
Some useful integral relations	45	(2)
Tensor analysis	47	(1)
Contravariant and covariant vectors	48	(1)
Tensors of second rank	48	(1)
Basic operations with tensors	49	(1)
Quotient law	50	(1)
The line element and metric tensor	51	(2)
Associated tensors	53	(1)
Geodesics in a Riemannian space	53	(2)
Covariant differentiation	55	(2)
Problems	57	(5)
Ordinary differential equations	62	(38)
First-order differential equations	63	(9)
Separable variables	63	(4)
Exact equations	67	(2)
Integrating factors	69	(3)
Bernoulli's equation	72	(1)
Second-order equations with constant coefficients	72	(11)
Nature of the solution of linear equations	73	(1)
General solutions of the second-order equations	74	(1)
Finding the complementary function	74	(3)
Finding the particular integral	77	(1)
Particular integral and the operator D(= d / dx)	78	(1)
Rules for D operators	79	(4)
The Euler linear equation	83	(2)
Solutions in power series	85	(8)
Ordinary and singular points of a differential equation	86	(1)
Frobenius and Fuchs theorem	86	(7)
Simultaneous equations	93	(1)
The gamma and beta functions	94	(2)
Problems	96	(4)
Matrix algebra	100	(44)
Definition of a matrix	100	(2)
Four basic algebra operations for matrices	102	(5)
Equality of matrices	102	(1)
Addition of matrices	102	(1)
Multiplication of a matrix by a number	103	(1)
Matrix multiplication	103	(4)
The commutator	107	(1)
Powers of a matrix	107	(1)
Functions of matrices	107	(1)
Transpose of a matrix	108	(1)
Symmetric and skew-symmetric matrices	109	(1)
The matrix representation of a vector product	110	(1)
The inverse of a matrix	111	(1)
A method for finding A-1	112	(1)
Systems of linear equations and the inverse of a matrix	113	(1)
Complex conjugate of a matrix	114	(1)
Hermitian conjugation	114	(1)
Hermitian/anti-hermitian matrix	114	(1)
Orthogonal matrix (real)	115	(1)
Unitary matrix	116	(1)
Rotation matrices	117	(4)
Trace of a matrix	121	(1)
Orthogonal and unitary transformations	121	(1)
Similarity transformation	122	(2)
The matrix eigenvalue problem	124	(4)
Determination of eigenvalues and eigenvectors	124	(4)
Eigenvalues and eigenvectors of hermitian matrices	128	(1)
Diagonalization of a matrix	129	(4)
Eigenvectors of commuting matrices	133	(1)
Cayley-Hamilton theorem	134	(1)
Moment of inertia matrix	135	(1)
Normal modes of vibrations	136	(3)
Direct product of matrices	139	(1)
Problems	140	(4)
Fourier series and integrals	144	(55)
Periodic functions	144	(2)
Fourier series; Euler-Fourier formulas	146	(4)
Gibb's phenomena	150	(1)
Convergence of Fourier series and Dirichlet conditions	150	(1)
Half-range Fourier series	151	(1)
Change of interval	152	(1)
Parseval's identity	153	(2)
Alternative forms of Fourier series	155	(2)
Integration and differentiation of a Fourier series	157	(1)
Vibrating strings	157	(3)
The equation of motion of transverse vibration	157	(1)
Solution of the wave equation	158	(2)
RLC circuit	160	(2)
Orthogonal functions	162	(1)
Multiple Fourier series	163	(1)
Fourier integrals and Fourier transforms	164	(8)
Fourier sine and cosine transforms	172	(1)
Heisenberg's uncertainty principle	173	(1)
Wave packets and group velocity	174	(5)
Heat conduction	179	(3)
Heat conduction equation	179	(3)
Fourier transforms for functions of several variables	182	(1)
The Fourier integral and the delta function	183	(3)
Parseval's identity for Fourier integrals	186	(2)
The convolution theorem for Fourier transforms	188	(2)
Calculations of Fourier transforms	190	(2)
The delta function and Green's function method	192	(3)
Problems	195	(4)
Linear vector spaces	199	(34)
Euclidean n-space En	199	(2)
General linear vector spaces	201	(2)
Subspaces	203	(1)
Linear combination	204	(1)
Linear independence, bases, and dimensionality	204	(2)
Inner product spaces (unitary spaces)	206	(3)
The Gram-Schmidt orthogonalization process	209	(1)
The Cauchy-Schwarz inequality	210	(1)
Dual vectors and dual spaces	211	(1)
Linear operators	212	(2)
Matrix representation of operators	214	(1)
The algebra of linear operators	215	(2)
Eigenvalues and eigenvectors of an operator	217	(1)
Some special operators	217	(7)
The inverse of an operator	218	(1)
The adjoint operators	219	(1)
Hermitian operators	220	(1)
Unitary operators	221	(1)
The projection operators	222	(2)
Change of basis	224	(1)
Commuting operators	225	(1)
Function spaces	226	(4)
Problems	230	(3)
Functions of a complex variable	233	(63)
Complex numbers	233	(5)
Basic operations with complex numbers	234	(1)
Polar form of complex number	234	(3)
De Moivre's theorem and roots of complex numbers	237	(1)
Functions of a complex variable	238	(1)
Mapping	239	(1)
Branch lines and Riemann surfaces	240	(1)
The differential calculus of functions of a complex variable	241	(8)
Limits and continuity	241	(2)
Derivatives and analytic functions	243	(1)
The Cauchy-Riemann conditions	244	(3)
Harmonic functions	247	(1)
Singular points	248	(1)
Elementary functions of z	249	(5)
The exponential functions ez (or exp(z)	249	(2)
Trigonometric and hyperbolic functions	251	(1)
The logarithmic functions w = In z	252	(1)
Hyperbolic functions	253	(1)
Complex integration	254	(11)
Line integrals in the complex plane	254	(3)
Cauchy's integral theorem	257	(3)
Cauchy's integral formulas	260	(2)
Cauchy's integral formulas for higher derivatives	262	(3)
Series representations of analytic functions	265	(14)
Complex sequences	265	(1)
Complex series	266	(2)
Ratio test	268	(1)
Uniform covergence and the Weierstrass M-test	268	(1)
Power series and Taylor series	269	(3)
Taylor series of elementary functions	272	(2)
Laurent series	274	(5)
Integration by the method of residues	279	(4)
Residues	279	(3)
The residue theorem	282	(1)
Evaluation of real definite integrals	283	(9)
Improper integrals of the rational function ∫∞-∞ƒ(x)dx	283	(3)
Integrals of the rational functions of sin &thetas; and cos &thetas; ∫2π0 G(sin &thetas;, cos &thetas;)d&thetas;	286	(2)
Fourier integrals of the form ∫∞-∞ƒ(x){sin mx cos mx}dx	288	(4)
Problems	292	(4)
Special functions of mathematical physics	296	(51)
Legendre's equation	296	(11)
Rodrigues' formula for Pn(x)	299	(2)
The generating function for Pn(x)	301	(3)
Orthogonality of Legendre polynomials	304	(3)
The associated Legendre functions	307	(4)
Orthogonality of associated Legendre functions	309	(2)
Hermite's equation	311	(5)
Rodrigues' formula for Hermite polynomials Hn(x)	313	(1)
Recurrence relations for Hermite polynomials	313	(1)
Generating function for the Hn(x)	314	(1)
The orthogonal Hermite functions	314	(2)
Laguerre's equation	316	(4)
The generating function for the Laguerre polynomials Ln(x)	317	(1)
Rodrigues' formula for the Laguerre polynomials Ln(x)	318	(1)
The orthogonal Laugerre functions	319	(1)
The associated Laguerre polynomials Lmn(x)	320	(1)
Generating function for the asociated Laguerre polynomials	320	(1)
Associated Laguerre function of integral order	321	(1)
Bessel's equation	321	(17)
Bessel functions of the second kind Yn(x)	325	(3)
Hanging flexible chain	328	(2)
Generating function for Jn(x)	330	(1)
Bessel's integral representation	331	(1)
Recurrence formulas for Jn(x)	332	(3)
Approximations to the Bessel functions	335	(1)
Orthogonality of Bessel functions	336	(2)
Spherical Bessel functions	338	(2)
Sturm-Liouville systems	340	(3)
Problems	343	(4)
The calculus of variations	347	(25)
The Euler-Lagrange equation	348	(5)
Variational problems with constraints	353	(2)
Hamilton's principle and Lagrange's equation of motion	355	(4)
Rayleigh-Ritz method	359	(2)
Hamilton's principle and canonical equations of motion	361	(3)
The modified Hamilton's principle and the Hamilton-Jacobi equation	364	(3)
Variational problems with several independent variables	367	(2)
Problems	369	(3)
The Laplace transformation	372	(15)
Definition of the Lapace transform	372	(1)
Existence of Laplace transforms	373	(2)
Laplace transforms of some elementary functions	375	(3)
Shifting (or translation) theorems	378	(2)
The first shifting theorem	378	(1)
The second shifting theorem	379	(1)
The unit step function	380	(1)
Laplace transform of a periodic function	381	(1)
Laplace transforms of derivatives	382	(1)
Laplace transforms of functions defined by integrals	383	(1)
A note on integral transformations	384	(1)
Problems	385	(2)
Partial differential equations	387	(26)
Linear second-order partial differential equations	388	(4)
Solutions of Laplace's equation: separation of variables	392	(10)
Solutions of the wave equation: separation of variables	402	(2)
Solution of Poisson's equation. Green's functions	404	(5)
Laplace transform solutions of boundary-value problems	409	(1)
Problems	410	(3)
Simple linear integral equations	413	(17)
Classification of linear integral equations	413	(1)
Some methods of solution	414	(7)
Separable kernel	414	(2)
Neumann series solutions	416	(3)
Transformation of an integral equation into a differential equation	419	(1)
Laplace transform solution	420	(1)
Fourier transform solution	421	(1)
The Schmidt-Hilbert method of solution	421	(4)
Relation between differential and integral equations	425	(1)
Use of integral equations	426	(2)
Abel's integral equation	426	(1)
Classical simple harmonic oscillator	427	(1)
Quantum simple harmonic oscillator	427	(1)
Problems	428	(2)
Elements of group theory	430	(29)
Definition of a group (group axioms)	430	(3)
Cyclic groups	433	(1)
Group multiplication table	434	(1)
Isomorphic groups	435	(3)
Group of permutations and Cayley's theorem	438	(1)
Subgroups and cosets	439	(1)
Conjugate classes and invariant subgroups	440	(2)
Group representations	442	(2)
Some special groups	444	(13)
The symmetry group D2, D3	446	(3)
One-dimensional unitary group U(1)	449	(1)
Orthogonal groups SO(2) and SO(3)	450	(2)
The SU(n) groups	452	(2)
Homogeneous Lorentz group	454	(3)
Problems	457	(2)
Numerical methods	459	(22)
Interpolation	459	(1)
Finding roots of equations	460	(1)
Graphical methods	460	(1)
Method of linear interpolation (method of false position)	461	(5)
Newton's method	464	(2)
Numerical integration	466	(3)
The rectangular rule	466	(1)
The trapezoidal rule	467	(2)
Simpson's rule	469	(1)
Numerical solutions of differential equations	469	(8)
Euler's method	470	(2)
The three-term Taylor series method	472	(1)
The Runge-Kutta method	473	(3)
Equations of higher order. System of equations	476	(1)
Least-squares fit	477	(1)
Problems	478	(3)
Introduction to probability theory	481	(25)
A definition of probability	481	(1)
Sample space	482	(2)
Methods of counting	484	(2)
Permutations	484	(1)
Combinations	485	(1)
Fundamental probability theorems	486	(3)
Random variables and probability distributions	489	(2)
Random variables	489	(1)
Probability distributions	489	(1)
Expectation and variance	490	(1)
Special probability distributions	491	(9)
The binomial distribution	491	(4)
The Poisson distribution	495	(2)
The Gaussian (or normal) distribution	497	(3)
Continuous distributions	500	(3)
The Gaussian (or normal) distribution	502	(1)
The Maxwell-Boltzmann distribution	503	(1)
Problems	503	(3)
Appendix 1 Preliminaries (review of fundamental concepts)	506	(32)
Inequalities	507	(1)
Functions	508	(2)
Limits	510	(1)
Infinite series	511	(9)
Tests for convergence	513	(3)
Alternating series test	516	(1)
Absolute and conditional convergence	517	(3)
Series of functions and uniform convergence	520	(4)
Weistrass M test	521	(1)
Abel's test	522	(2)
Theorem on power series	524	(1)
Taylor's expansion	524	(4)
Higher derivatives and Leibnitz's formula for nth derivative of a product	528	(1)
Some important properties of definite integrals	529	(2)
Some useful methods of integration	531	(2)
Reduction formula	533	(1)
Differentiation of integrals	534	(1)
Homogeneous functions	535	(1)
Taylor series for functions of two independent variables	535	(1)
Lagrange multiplier	536	(2)
Appendix 2 Determinants	538	(10)
Determinants, minors, and cofactors	540	(1)
Expansion of determinants	541	(1)
Properties of determinants	542	(5)
Derivative of a determinant	547	(1)
Appendix 3 Table of function F(x) = 1-√2π ∫x0 e-t2/2 dt	548	(1)
Further reading	549	(2)
Index	551