Mathematical Methods for Physics and Engineering: A Comprehensive Guide

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

• ISBN13: 9780521861533
• ISBN10: 0521861535
• Edition: 3rd
• Format: Hardcover
• Copyright: 2006-03-20
• 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	1	(9)
Polynomial equations; factorisation; properties of roots
Trigonometric identities	10	(5)
Single angle; compound angles; double- and half-angle identities
Coordinate geometry	15	(3)
Partial fractions	18	(7)
Complications and special cases
Binomial expansion	25	(2)
Properties of binomial coefficients	27	(3)
Some particular methods of proof	30	(6)
Proof by induction; proof by contradiction; necessary and sufficient conditions
Exercises	36	(3)
Hints and answers	39	(2)
Preliminary calculus	41	(42)
Differentiation	41	(18)
Differentiation from first principles; products; the chain rule; quotients; implicit differentiation; logarithmic differentiation; Leibnitz' theorem; special points of a function; curvature; theorems of differentiation
Integration	59	(17)
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
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	85	(7)
Addition and subtraction; modulus and argument; multiplication; complex conjugate; division
Polar representation of complex numbers	92	(3)
Multiplication and division in polar form
de Moivre's theorem	95	(4)
trigonometric identities; finding the nth roots of unity; solving polynomial equations
Complex logarithms and complex powers	99	(2)
Applications to differentiation and integration	101	(1)
Hyperbolic functions	102	(7)
Definitions; hyperbolic-trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculus of hyperbolic functions
Exercises	109	(4)
Hints and answers	113	(2)
Series and limits	115	(36)
Series	115	(1)
Summation of series	116	(8)
Arithmetic series; geometric series; arithmetico-geometric series; the difference method; series involving natural numbers; transformation of series
Convergence of infinite series	124	(7)
Absolute and conditional convergence; series containing only real positive terms; alternating series test
Operations with series	131	(1)
Power series	131	(5)
Convergence of power series; operations with power series
Taylor series	136	(5)
Taylor's theorem; approximation errors; standard Maclaurin series
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	191	(8)
Areas and volumes; masses, centres of mass and centroids; Pappus' theorems; moments of inertia; mean values of functions
Change of variables in multiple integrals	199	(8)
Change of variables in double integrals; evaluation of the integral I = dx; change of variables in triple integrals; general properties of Jacobians
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	219	(7)
Scalar product; vector product; scalar triple product; vector triple product
Equations of lines, planes and spheres	226	(3)
Using vectors to find distances	229	(4)
Point to line; point to plane; line to line; line to plane
Reciprocal vectors	233	(1)
Exercises	234	(6)
Hints and answers	240	(1)
Matrices and vector spaces	241	(75)
Vector spaces	242	(5)
Basis vectors; inner product; some useful inequalities
Linear operators	247	(2)
Matrices	249	(1)
Basic matrix algebra	250	(5)
Matrix addition; multiplication by a scalar; matrix multiplication
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	259	(4)
Properties of determinants
The inverse of a matrix	263	(4)
The rank of a matrix	267	(1)
Special types of square matrix	268	(4)
Diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian and anti-Hermitian; unitary; normal
Eigenvectors and eigenvalues	272	(8)
Of a normal matrix; of Hermitian and anti-Hermitian matrices; of a unitary matrix; of a general square matrix
Determination of eigenvalues and eigenvectors	280	(2)
Degenerate eigenvalues
Change of basis and similarity transformations	282	(3)
Diagonalisation of matrices	285	(3)
Quadratic and Hermitian forms	288	(4)
Stationary properties of the eigenvectors; quadratic surfaces
Simultaneous linear equations	292	(15)
Range; null space; N simultaneous linear equations in N unknowns; singular value decomposition
Exercises	307	(7)
Hints and answers	314	(2)
Normal modes	316	(18)
Typical oscillatory systems	317	(5)
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	334	(5)
Composite vector expressions; differential of a vector
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	347	(7)
Gradient of a scalar field; divergence of a vector field; curl of a vector field
Vector operator formulae	354	(3)
Vector operators acting on sums and products; combinations of grad, div and curl
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	377	(6)
Evaluating line integrals; physical examples; line integrals with respect to a scalar
Connectivity of regions	383	(1)
Green's theorem in a plane	384	(3)
Conservative fields and potentials	387	(2)
Surface integrals	389	(7)
Evaluating surface integrals; vector areas of surfaces; physical examples
Volume integrals	396	(2)
Volumes of three-dimensional regions
Integral forms for grad, div and curl	398	(3)
Divergence theorem and related theorems	401	(5)
Green's theorems; other related integral theorems; physical applications
Stokes' theorem and related theorems	406	(3)
Related integral theorems; physical applications
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	(98)
Fourier transforms	433	(20)
The uncertainty principle; Fraunhofer diffraction; the Dirac δ-function; relation of the δ-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
Laplace transforms	453	(6)
Laplace transforms of derivatives and integrals; other properties of Laplace transforms
Concluding remarks	459	(1)
Exercises	460	(6)
Hints and answers	466	(2)
First-order ordinary differential equations	468	(1)
General form of solution	469	(1)
First-degree first-order equations	470	(10)
Separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; isobaric equations; Bernoulli's equation; miscellaneous equations
Higher-degree first-order equations	480	(4)
Equations soluble for p; for x; for y; Clairaut's equation
Exercises	484	(4)
Hints and answers	488	(2)
Higher-order ordinary differential equations	490	(2)
Linear equations with constant coefficients	492	(11)
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
Linear equations with variable coefficients	503	(15)
The Legendre and Euler linear equations; exact equations; partially known complementary function; variation of parameters; Green's functions; canonical form for second-order equations
General ordinary differential equations	518	(5)
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
Exercises	523	(6)
Hints and answers	529	(2)
Series solutions of ordinary differential equations	531	(23)
Second-order linear ordinary differential equations	531	(4)
Ordinary and singular points
Series solutions about an ordinary point	535	(3)
Series solutions about a regular singular point	538	(6)
Distinct roots not differing by an integer; repeated root of the indicial equation; distinct roots differing by an integer
Obtaining a second solution	544	(4)
The Wronskian method; the derivative method; series form of the second solution
Polynomial solutions	548	(2)
Exercises	550	(3)
Hints and answers	553	(1)
Eigenfunction methods for differential equations	554	(23)
Sets of functions	556	(3)
Some useful inequalities
Adjoint, self-adjoint and Hermitian operators	559	(2)
Properties of Hermitian operators	561	(3)
Reality of the eigenvalues; orthogonality of the eigenfunctions; construction of real eigenfunctions
Sturm--Liouville equations	564	(5)
Valid boundary conditions; putting an equation into Sturm--Liouville form
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	577	(10)
General solution for integer l; properties of Legendre polynomials
Associated Legendre functions	587	(6)
Spherical harmonics	593	(2)
Chebyshev functions	595	(7)
Bessel functions	602	(12)
General solution for non-integer v; general solution for integer v; properties of Bessel functions
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	648	(8)
Commutators
Physical examples of operators	656	(15)
Uncertainty principle; angular momentum; creation and annihilation operators
Exercises	671	(3)
Hints and answers	674	(1)
Partial differential equations: general and particular solutions	675	(38)
Important partial differential equations	676	(4)
The wave equation; the diffusion equation; Laplace's equation; Poisson's equation; Schrodinger's equation
General form of solution	680	(1)
General and particular solutions	681	(12)
First-order equations; inhomogeneous equations and problems; second-order equations
The wave equation	693	(2)
The diffusion equation	695	(4)
Characteristics and the existence of solutions	699	(6)
First-order equations; second-order equations
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	725	(22)
Laplace's equation in polar coordinates; spherical harmonics; other equations in polar coordinates; solution by expansion; separation of variables for inhomogeneous equations
Integral transform methods	747	(4)
Inhomogeneous problems -- Green's functions	751	(16)
Similarities to Green's functions for ordinary differential equations; general boundary-value problems; Dirichlet problems; Neumann problems
Exercises	767	(6)
Hints and answers	773	(2)
Calculus of variations	775	(28)
The Euler--Lagrange equation	776	(1)
Special cases	777	(4)
F does not contain y explicitly; F does not contain x explicitly
Some extensions	781	(4)
Several dependent variables; several independent variables; higher-order derivatives; variable end-points
Constrained variation	785	(2)
Physical variational principles	787	(3)
Fermat's principle in optics; Hamilton's principle in mechanics
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	806	(7)
Separable kernels; integral transform methods; differentiation
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)
WK B methods	895	(10)
Approximations to integrals	905	(15)
Level lines and saddle points; steepest descents; stationary phase
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 Christoffel symbols	965	(3)
Covariant differentiation	968	(3)
Vector operators in tensor form	971	(4)
Absolute derivatives along curves	975	(1)
Geodesics	976	(1)
Exercises	977	(5)
Hints and answers	982	(2)
Numerical methods	984	(57)
Algebraic and transcendental equations	985	(7)
Rearrangement of the equation; linear interpolation; binary chopping; Newton--Raphson method
Convergence of iteration schemes	992	(2)
Simultaneous linear equations	994	(6)
Gaussian elimination; Gauss--Seidel iteration; tridiagonal matrices
Numerical integration	1000	(19)
Trapezium rule; Simpson's rule; Gaussian integration; Monte Carlo methods
Finite differences	1019	(1)
Differential equations	1020	(8)
Difference equations; Taylor series solutions; prediction and correction; Runge--Kutta methods; isoclines
Higher-order equations	1028	(2)
Partial differential equations	1030	(3)
Exercises	1033	(6)
Hints and answers	1039	(2)
Group theory	1041	(35)
Groups	1041	(8)
Definition of a group; examples of groups
Finite groups	1049	(3)
Non-Abelian groups	1052	(4)
Permutation groups	1056	(3)
Mappings between groups	1059	(2)
Subgroups	1061	(2)
Subdividing a group	1063	(7)
Equivalence relations and classes; congruence and cosets; conjugates and classes
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	1092	(3)
Orthogonality property of characters
Counting irreps using characters	1095	(5)
Summation rules for irreps
Construction of a character table	1100	(2)
Group nomenclature	1102	(1)
Product representations	1103	(2)
Physical applications of group theory	1105	(8)
Bonding in molecules; matrix elements in quantum mechanics; degeneracy of normal modes; breaking of degeneracies
Exercises	1113	(4)
Hints and answers	1117	(2)
Probability	1119	(102)
Venn diagrams	1119	(5)
Probability	1124	(9)
Axioms and theorems; conditional probability; Bayes' theorem
Permutations and combinations	1133	(6)
Random variables and distributions	1139	(4)
Discrete random variables; continuous random variables
Properties of distributions	1143	(7)
Mean; mode and median; variance and standard deviation; moments; central moments
Functions of random variables	1150	(7)
Generating functions	1157	(11)
Probability generating functions; moment generating functions; characteristic functions; cumulant generating functions
Important discrete distributions	1168	(11)
Binomial; geometric; negative binomial; hypergeometric; Poisson
Important continuous distributions	1179	(16)
Gaussian; log-normal: exponential; gamma; chi-squared; Cauchy; Breit-Wigner; uniform
The central limit theorem	1195	(1)
Joint distributions	1196	(3)
Discrete bivariate; continuous bivariate; marginal and conditional distributions
Properties of joint distributions	1199	(6)
Means; variances; covariance and correlation
Generating functions for joint distributions	1205	(1)
Transformation of variables in joint distributions	1206	(1)
Important joint distributions	1207	(4)
Multinominal; multivariate Gaussian
Exercises	1211	(8)
Hints and answers	1219	(2)
Statistics	1221	(84)
Experiments, samples and populations	1221	(1)
Sample statistics	1222	(7)
Averages; variance and standard deviation; moments; covariance and correlation
Estimators and sampling distributions	1229	(14)
Consistency, bias and efficiency; Fisher's inequality; standard errors; confidence limits
Some basic estimators	1243	(12)
Mean; variance; standard deviation; moments; covariance and correlation
Maximum-likelihood method	1255	(16)
ML estimator; transformation invariance and bias; efficiency; errors and confidence limits; Bayesian interpretation; large-N behaviour; extended ML method
The method of least squares	1271	(6)
Linear least squares; non-linear least squares
Hypothesis testing	1277	(21)
Simple and composite hypotheses; statistical tests; Neyman--Pearson; generalised likelihood-ratio; Student's t; Fisher's F; goodness of fit
Exercises	1298	(5)
Hints and answers	1303	(2)
Index	1305