Mathematical Methods for Physics and Engineering: A Comprehensive Guide

by K. F. Riley , M. P. Hobson , S. J. Bence

• ISBN13: 9780521679718
• ISBN10: 0521679710
• Edition: 3rd
• Format: Paperback
• Copyright: 2006-03-13
• Publisher: Cambridge University Press

Summary

The third edition of this highly acclaimed undergraduate textbook is suitable for teaching all the mathematics for an undergraduate course in any of the physical sciences. As well as lucid descriptions of all the topics and many worked examples, it contains over 800 exercises. New stand-alone chapters give a systematic account of the 'special functions' of physical science, cover an extended range of practical applications of complex variables, and give an introduction to quantum operators. Further tabulations, of relevance in statistics and numerical integration, have been added. In this edition, half of the exercises are provided with hints and answers and, in a separate manual available to both students and their teachers, complete worked solutions. The remaining exercises have no hints, answers or worked solutions and can be used for unaided homework; full solutions are available to instructors on a password-protected web site, www.cambridge.org/9780521679718.

Table of Contents

Preface to the third edition	xx
Preface to the second edition	xxiii
Preface to the first edition	xxv
Preliminary algebra	1	(40)
Simple functions and equations
Polynomial equations: factorisation; properties of roots	1	(9)
Trigonometric identities
Single angle; compound angles; double- and half-angle identities	10	(5)
Coordinate geometry	15	(3)
Partial fractions
Complications and special cases	18	(7)
Binomial expansion	25	(2)
Properties of binomial coefficients	27	(3)
Some particular methods of proof
Proof by induction; proof by contradiction; necessary and sufficient conditions	30	(6)
Exercises	36	(3)
Hints and answers	39	(2)
Preliminary calculus	41	(42)
Differentiation
Differentiation from first principles; products; the chain rule; quotients; implicit differentiation; logarithmic differentiation; Leibnitz' theorem; special points of a function; curvature; theorems of differentiation	41	(18)
Integration
Integration from first principles: the inverse of differentiation; by inspection; sinusoidal functions; logarithmic integration; using partial fractions; substitution method; integration by parts; reduction formulae; infinite and improper integrals; plane polar coordinates; integral inequalities; applications of integration	59	(17)
Exercises	76	(5)
Hints and answers	81	(2)
Complex numbers and hyperbolic functions	83	(32)
The need for complex numbers	83	(2)
Manipulation of complex numbers
Addition and subtraction; modulus and argument; multiplication; complex conjugate; division	85	(7)
Polar representation of complex numbers
Multiplication and division in polar form	92	(3)
de Moivre's theorem
trigonometric identities; finding the nth roots of unity; solving polynomial equations	95	(4)
Complex logarithms and complex powers	99	(2)
Applications to differentiation and integration	101	(1)
Hyperbolic functions
Definitions; hyperbolic-trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculus of hyperbolic functions	102	(7)
Exercises	109	(4)
Hints and answers	113	(2)
Series and limits	115	(36)
Series	115	(1)
Summation of series
Arithmetic series; geometric series; arithmetico-geometric series; the difference method; series involving natural numbers; transformation of series	116	(8)
Convergence of infinite series
Absolute and conditional convergence; series containing only real positive terms; alternating series test	124	(7)
Operations with series	131	(1)
Power series
Convergence of power series; operations with power series	131	(5)
Taylor series
Taylor's theorem; approximation errors; standard Maclaurin series	136	(5)
Evaluation of limits	141	(3)
Exercises	144	(5)
Hints and answers	149	(2)
Partial differentiation	151	(36)
Definition of the partial derivative	151	(2)
The total differential and total derivative	153	(2)
Exact and inexact differentials	155	(2)
Useful theorems of partial differentiation	157	(1)
The chain rule	157	(1)
Change of variables	158	(2)
Taylor's theorem for many-variable functions	160	(2)
Stationary values of many-variable functions	162	(5)
Stationary values under constraints	167	(6)
Envelopes	173	(3)
Thermodynamic relations	176	(2)
Differentiation of integrals	178	(1)
Exercises	179	(6)
Hints and answers	185	(2)
Multiple integrals	187	(25)
Double integrals	187	(3)
Triple integrals	190	(1)
Applications of multiple integrals
Areas and volumes; masses, centres of mass and centroids; Pappus' theorems: moments of inertia; mean values of functions	191	(8)
Change of variables in multiple integrals
Change of variables in double integrals; evaluation of the integral I = ∞-∞ e-x2 change of variables in triple integrals; general properties of Jacobians	199	(8)
Exercises	207	(4)
Hints and answers	211	(1)
Vector algebra	212	(29)
Scalars and vectors	212	(1)
Addition and subtraction of vectors	213	(1)
Multiplication by a scalar	214	(3)
Basis vectors and components	217	(1)
Magnitude of a vector	218	(1)
Multiplication of vectors
Scalar product; vector product; scalar triple product; vector triple product	219	(7)
Equations of lines, planes and spheres	226	(3)
Using vectors to find distances
Point to line; point to plane; line to line; line to plane	229	(4)
Reciprocal vectors	233	(1)
Exercises	234	(6)
Hints and answers	240	(1)
Matrices and vector spaces	241	(75)
Vector spaces
Basis vectors; inner product; some useful inequalities	242	(5)
Linear operators	247	(2)
Matrices	249	(1)
Basic matrix algebra
Matrix addition; multiplication by a scalar; matrix multiplication	250	(5)
Functions of matrices	255	(1)
The transpose of a matrix	255	(1)
The complex and Hermitian conjugates of a matrix	256	(2)
The trace of a matrix	258	(1)
The determinant of a matrix
Properties of determinants	259	(4)
The inverse of a matrix	263	(4)
The rank of a matrix	267	(1)
Special types of square matrix
Diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian and anti-Hermitian; unitary; normal	268	(4)
Eigenvectors and eigenvalues
Of a normal matrix; of Hermitian and anti-Hermitian matrices; of a unitary matrix; of a general square matrix	272	(8)
Determination of eigenvalues and eigenvectors
Degenerate eigenvalues	280	(2)
Change of basis and similarity transformations	282	(3)
Diagonalisation of matrices	285	(3)
Quadratic and Hermitian forms
Stationary properties of the eigenvectors; quadratic surfaces	288	(4)
Simultaneous linear equations
Range; null space; N simultaneous linear equations in N unknowns; singular value decomposition	292	(15)
Exercises	307	(7)
Hints and answers	314	(2)
Normal modes	316	(18)
Typical oscillatory systems	31	(291)
Symmetry and normal modes	322	(5)
Rayleigh-Ritz method	327	(2)
Exercises	329	(3)
Hints and answers	332	(2)
Vector calculus	334	(43)
Differentiation of vectors
Composite vector expressions; differential of a vector	334	(5)
Integration of vectors	339	(1)
Space curves	340	(4)
Vector functions of several arguments	344	(1)
Surfaces	345	(2)
Scalar and vector fields	347	(1)
Vector operators
Gradient of a scalar field; divergence of a vector field; curl of a vector field	347	(7)
Vector operator formulae
Vector operators acting on sums and products; combinations of grad, div and curl	354	(3)
Cylindrical and spherical polar coordinates	357	(7)
General curvilinear coordinates	364	(5)
Exercises	369	(6)
Hints and answers	375	(2)
Line, surface and volume integrals	377	(38)
Line integrals
Evaluating line integrals; physical examples; line integrals with respect to a scalar	377	(6)
Connectivity of regions	383	(1)
Green's theorem in a plane	384	(3)
Conservative fields and potentials	387	(2)
Surface integrals
Evaluating surface integrals; vector areas of surfaces; physical examples	389	(7)
Volume integrals
Volumes of three-dimensional regions	396	(2)
Integral forms for grad, div and curl	398	(3)
Divergence theorem and related theorems
Green's theorems; other related integral theorems; physical applications	401	(5)
Stokes' theorem and related theorems
Related integral theorems; physical applications	406	(3)
Exercises	409	(5)
Hints and answers	414	(1)
Fourier series	415	(18)
The Dirichlet conditions	415	(2)
The Fourier coefficients	417	(2)
Symmetry considerations	419	(1)
Discontinuous functions	420	(2)
Non-periodic functions	422	(2)
Integration and differentiation	424	(1)
Complex Fourier series	424	(2)
Parseval's theorem	426	(1)
Exercises	427	(4)
Hints and answers	431	(2)
Integral transforms	433	(35)
Fourier transforms
The uncertainty principle; Fraunhofer diffraction; the Dirac o-function; relation of the o-function to Fourier transforms; properties of Fourier transforms; odd and even functions; convolution and deconvolution; correlation functions and energy spectra; Parseval's theorem; Fourier transforms in higher dimensions	433	(20)
Laplace transforms
Laplace transforms of derivatives and integrals; other properties of Laplace transforms	453	(6)
Concluding remarks	459	(1)
Exercises	460	(6)
Hints and answers	466	(2)
First-order ordinary differential equations	468	(22)
General form of solution	469	(1)
First-degree first-order equations
Separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; isobaric equations; Bernoulli's equation; miscellaneous equations	470	(10)
Higher-degree first-order equations
Equations soluble for p; for x; for y; Clairaut's equation	480	(4)
Exercises	484	(4)
Hints and answers	488	(2)
Higher-order ordinary differential equations	490	(41)
Linear equations with constant coefficients
Finding the complementary function yc(x); finding the particular integral yp(x); constructing the general solution yc(x) + yp(x); linear recurrence relations; Laplace transform method	492	(11)
Linear equations with variable coefficients
The Legendre and Euler linear equations; exact equations; partially known complementary function; variation of parameters; Green's functions; canonical form for second-order equations	503	(15)
General ordinary differential equations
Dependent variable absent: independent variable absent; non-linear exact equations: isobaric or homogeneous equations: equations homogeneous in x or y alone; equations having y = Aex as a solution	518	(5)
Exercises	523	(6)
Hints and answers	529	(2)
Series solutions of ordinary differential equations	531	(23)
Second-order linear ordinary differential equations
Ordinary and singular points	531	(4)
Series solutions about an ordinary point	535	(3)
Series solutions about a regular singular point
Distinct roots not differing by an integer; repeated root of the indicial equation; distinct roots differing by an integer	538	(6)
Obtaining a second solution
The Wronskian method; the derivative method; series form of the second solution	544	(4)
Polynomial solutions	548	(2)
Exercises	550	(3)
Hints and answers	553	(1)
Eigenfunction methods for differential equations	554	(23)
Sets of functions
Some useful inequalities	556	(3)
Adjoint, self-adjoint and Hermitian operators	559	(2)
Properties of Hermitian operators
Reality of the eigenvalues; orthogonality of the eigenfunctions; construction of real eigenfunctions	561	(3)
Sturm-Liouville equations
Valid boundary conditions; putting an equation into Sturm-Liouville form	564	(5)
Superposition of eigenfunctions: Green's functions	569	(3)
A useful generalisation	572	(1)
Exercises	573	(3)
Hints and answers	576	(1)
Special functions	577	(71)
Legendre functions
General solution for integer f; properties of Legendre polynomials	577	(10)
Associated Legendre functions	587	(6)
Spherical harmonics	593	(2)
Chebyshev functions	595	(7)
Bessel functions
General solution for non-integer v; general solution for integer v; properties of Bessel functions	602	(12)
Spherical Bessel functions	614	(2)
Laguerre functions	616	(5)
Associated Laguerre functions	621	(3)
Hermite functions	624	(4)
Hypergeometric functions	628	(5)
Confluent hypergeometric functions	633	(2)
The gamma function and related functions	635	(5)
Exercises	640	(6)
Hints and answers	646	(2)
Quantum operators	648	(27)
Operator formalism
Commutators	648	(8)
Physical examples of operators
Uncertainty principle: angular momentum; creation and annihilation operators	656	(15)
Exercises	671	(3)
Hints and answers	674	(1)
Partial differential equations: general and particular solutions	675	(38)
Important partial differential equations
The wave equation: the diffusion equation: Laplace's equation; Poisson's equation; Schrodinger's equation	676	(4)
General form of solution	680	(1)
General and particular solutions
First-order equations; inhomogeneous equations and problems; second-order equations	681	(12)
The wave equation	693	(2)
The diffusion equation	695	(4)
Characteristics and the existence of solutions
First-order equations; second-order equations	699	(6)
Uniqueness of solutions	705	(2)
Exercises	707	(4)
Hints and answers	711	(2)
Partial differential equations: separation of variables and other methods	713	(62)
Separation of variables: the general method	713	(4)
Superposition of separated solutions	717	(8)
Separation of variables in polar coordinates
Laplace's equation in polar coordinates; spherical harmonics; other equations in polar coordinates; solution by expansion; separation of variables for inhomogeneous equations	725	(22)
Integral transform methods	747	(4)
Inhomogeneous problems - Green's functions
Similarities to Green's functions for ordinary differential equations: general boundary-value problems; Dirichlet problems; Neumann problems	751	(16)
Exercises	767	(6)
Hints and answers	773	(2)
Calculus of variations	775	(28)
The Euler-Lagrange equation	776	(1)
Special cases
F does not contain y explicitly; F does not contain x explicitly	777	(4)
Some extensions
Several dependent variables; several independent variables; higher-order derivatives; variable end-points	781	(4)
Constrained variation	785	(2)
Physical variational principles
Fermat's principle in optics; Hamilton's principle in mechanics	787	(3)
General eigenvalue problems	790	(2)
Estimation of eigenvalues and eigenfunctions	792	(3)
Adjustment of parameters	795	(2)
Exercises	797	(4)
Hints and answers	801	(2)
Integral equations	803	(21)
Obtaining an integral equation from a differential equation	803	(1)
Types of integral equation	804	(1)
Operator notation and the existence of solutions	805	(1)
Closed-form solutions
Separable kernels; integral transform methods; differentiation	806	(7)
Neumann series	813	(2)
Fredholm theory	815	(1)
Schmidt-Hilbert theory	816	(3)
Exercises	819	(4)
Hints and answers	823	(1)
Complex variables	824	(47)
Functions of a complex variable	825	(2)
The Cauchy-Riemann relations	827	(3)
Power series in a complex variable	830	(2)
Some elementary functions	832	(3)
Multivalued functions and branch cuts	835	(2)
Singularities and zeros of complex functions	837	(2)
Conformal transformations	839	(6)
Complex integrals	845	(4)
Cauchy's theorem	849	(2)
Cauchy's integral formula	851	(2)
Taylor and Laurent series	853	(5)
Residue theorem	858	(3)
Definite integrals using contour integration	861	(6)
Exercises	867	(3)
Hints and answers	870	(1)
Applications of complex variables	871	(56)
Complex potentials	871	(5)
Applications of conformal transformations	876	(3)
Location of zeros	879	(3)
Summation of series	882	(2)
Inverse Laplace transform	884	(4)
Stokes' equation and Airy integrals	888	(7)
WKB methods	895	(10)
Approximations to integrals
Level lines and saddle points; steepest descents; stationary phase	905	(15)
Exercises	920	(5)
Hints and answers	925	(2)
Tensors	927	(57)
Some notation	928	(1)
Change of basis	929	(1)
Cartesian tensors	930	(2)
First- and zero-order Cartesian tensors	932	(3)
Second- and higher-order Cartesian tensors	935	(3)
The algebra of tensors	938	(1)
The quotient law	939	(2)
The tensors δij and εijk	941	(3)
Isotropic tensors	944	(2)
Improper rotations and pseudotensors	946	(3)
Dual tensors	949	(1)
Physical applications of tensors	950	(4)
Integral theorems for tensors	954	(1)
Non-Cartesian coordinates	955	(2)
The metric tensor	957	(3)
General coordinate transformations and tensors	960	(3)
Relative tensors	963	(2)
Derivatives of basis vectors and Christofiel symbols	965	(3)
Covariant differentiation	968	(3)
Vector operators in tensor form	971	(4)
Absolute derivatives along curves	975	(1)
Geodesies	976	(1)
Exercises	977	(5)
Hints and answers	982	(2)
Numerical methods	984	(57)
Algebraic and transcendental equations
Rearrangement of the equation: linear interpolation: binary chopping; Newton-Raphson method	985	(7)
Convergence of iteration schemes	992	(2)
Simultaneous linear equations
Gaussian elimination; Gauss-Seidel iteration; tridiagonal matrices	994	(6)
Numerical integration
Trapezium rule; Simpson's rule; Gaussian integration; Monte Carlo methods	1000	(19)
Finite differences	1019	(1)
Differential equations
Difference equations; Taylor series solutions; prediction and correction; Runge-Kutta methods; isoclines	1020	(8)
Higher-order equations	1028	(2)
Partial differential equations	1030	(3)
Exercises	1033	(6)
Hints and answers	1039	(2)
Group theory	1041	(35)
Groups
Definition of a group; examples of groups	1041	(8)
Finite groups	1049	(3)
Non-Abelian groups	1052	(4)
Permutation groups	1056	(3)
Mappings between groups	1059	(2)
Subgroups	1061	(2)
Subdividing a group
Equivalence relations and classes; congruence and cosets: conjugates and classes	1063	(7)
Exercises	1070	(4)
Hints and answers	1074	(2)
Representation theory	1076	(43)
Dipole moments of molecules	1077	(1)
Choosing an appropriate formalism	1078	(6)
Equivalent representations	1084	(2)
Reducibility of a representation	1086	(4)
The orthogonality theorem for irreducible representations	1090	(2)
Characters
Orthogonality property of characters	1092	(3)
Counting irreps using characters
Summation rules for irreps	1095	(5)
Construction of a character table	1100	(2)
Group nomenclature	1102	(1)
Product representations	1103	(2)
Physical applications of group theory
Bonding in molecules; matrix elements in quantum mechanics; degeneracy of normal modes; breaking of degeneracies	1105	(8)
Exercises	1113	(4)
Hints and answers	1117	(2)
Probability	1119	(102)
Venn diagrams	1119	(5)
Probability
Axioms and theorems; conditional probability; Bayes' theorem	1124	(9)
Permutations and combinations	1133	(6)
Random variables and distributions
Discrete random variables; continuous random variables	1139	(4)
Properties of distributions
Mean; mode and median; variance and standard deviation; moments; central moments	1143	(7)
Functions of random variables	1150	(7)
Generating functions
Probability generating functions; moment generating functions; characteristic functions; cumulant generating functions	1157	(11)
Important discrete distributions
Binomial; geometric; negative binomial; hypergeometric; Poisson	1168	(11)
Important continuous distributions
Gaussian; log-normal; exponential; gamma; chi-squared; Cauchy; Breit-Wigner; uniform	1179	(16)
The central limit theorem	1195	(1)
Joint distributions
Discrete bivariate; continuous bivariate; marginal and conditional distributions	1196	(3)
Properties of joint distributions
Means; variances; covariance and correlation	1199	(6)
Generating functions for joint distributions	1205	(1)
Transformation of variables in joint distributions	1206	(1)
Important joint distributions
Multinominal; multivariate Gaussian	1207	(4)
Exercises	1211	(8)
Hints and answers	1219	(2)
Statistics	1221	(84)
Experiments, samples and populations	1221	(1)
Sample statistics
Averages: variance and standard deviation; moments; covariance and correlation	1222	(7)
Estimators and sampling distributions
Consistency, bias and efficiency; Fisher's inequality; standard errors; confidence limits	1229	(14)
Some basic estimators
Mean; variance; standard deviation; moments; covariance and correlation	1243	(12)
Maximum-likelihood method
ML estimator; transformation invariance and bias; efficiency; errors and confidence limits; Bayesian interpretation; large-N behaviour; extended ML method	1255	(16)
The method of least squares
Linear least squares; non-linear least squares	1271	(6)
Hypothesis testing
Simple and composite hypotheses; statistical tests; Neyman-Pearson; generalised likelihood-ratio; Student's t; Fisher's F; goodness of fit	1277	(21)
Exercises	1298	(5)
Hints and answers	1303	(2)
Index	1305