Mathematical Methods for Physics and Engineering: A Comprehensive Guide

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

• ISBN13: 9780521890670
• ISBN10: 0521890675
• Edition: 2nd
• Format: Paperback
• Copyright: 2002-08-26
• Publisher: Cambridge University Press

Summary

The new edition of this highly acclaimed textbook contains several major additions, including more than four hundred new exercises (with hints and answers). To match the mathematical preparation of current senior college and university entrants, the authors have included a preliminary chapter covering areas such as polynomial equations, trigonometric identities, coordinate geometry, partial fractions, binomial expansions, induction, and the proof of necessary and sufficient conditions. Elsewhere, matrix decompositions, nearly-singular matrices and non-square sets of linear equations are treated in detail. The presentation of probability has been reorganized and greatly extended, and includes all physically important distributions. New topics covered in a separate statistics chapter include estimator efficiency, distributions of samples, t- and F- tests for comparing means and variances, applications of the chi-squared distribution, and maximum likelihood and least-squares fitting. In other chapters the following topics have been added: linear recurrence relations, curvature, envelopes, curve-sketching, and more refined numerical methods.

Table of Contents

Preface to the second edition	xix
Preface to the first edition	xxi
Preliminary algebra	1	(41)
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	(3)
Preliminary calculus	42	(44)
Differentiation	42	(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	60	(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	77	(5)
Hints and answers	82	(4)
Complex numbers and hyperbolic functions	86	(32)
The need for complex numbers	86	(2)
Manipulation of complex numbers	88	(7)
Addition and subtraction; modulus and argument; multiplication; complex conjugate; division
Polar representation of complex numbers	95	(3)
Multiplication and division in polar form
de Moivre's theorem	98	(4)
trigonometric identities; finding the nth roots of unity; solving polynomial equations
Complex logarithms and complex powers	102	(2)
Applications to differentiation and integration	104	(1)
Hyperbolic functions	105	(7)
Definitions; hyperbolic-trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculus of hyperbolic functions
Exercises	112	(4)
Hints and answers	116	(2)
Series and limits	118	(36)
Series	118	(1)
Summation of series	119	(8)
Arithmetic series; geometric series; arithmetico-geometric series; the difference method; series involving natural numbers; transformation of series
Convergence of infinite series	127	(7)
Absolute and conditional convergence; series containing only real positive terms; alternating series test
Operations with series	134	(1)
Power series	134	(5)
Convergence of power series; operations with power series
Taylor series	139	(5)
Taylor's theorem; approximation errors; standard Maclaurin series
Evaluation of limits	144	(3)
Exercises	147	(5)
Hints and answers	152	(2)
Partial differentiation	154	(36)
Definition of the partial derivative	154	(2)
The total differential and total derivative	156	(2)
Exact and inexact differentials	158	(2)
Useful theorems of partial differentiation	160	(1)
The chain rule	160	(1)
Change of variables	161	(2)
Taylor's theorem for many-variable functions	163	(2)
Stationary values of many-variable functions	165	(5)
Stationary values under constraints	170	(6)
Envelopes	176	(3)
Thermodynamic relations	179	(2)
Differentiation of integrals	181	(1)
Exercises	182	(6)
Hints and answers	188	(2)
Multiple integrals	190	(26)
Double integrals	190	(3)
Triple integrals	193	(1)
Applications of multiple integrals	194	(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	202	(8)
Change of variables in double integrals; evaluation of the integral I = ∫∞-∞ e-x2 dx; change of variables in triple integrals; general properties of Jacobians
Exercises	210	(4)
Hints and answers	214	(2)
Vector algebra	216	(30)
Scalars and vectors	216	(1)
Addition and subtraction of vectors	217	(1)
Multiplication by a scalar	218	(3)
Basis vectors and components	221	(1)
Magnitude of a vector	222	(1)
Multiplication of vectors	223	(7)
Scalar product; vector product; scalar triple product; vector triple product
Equations of lines, planes and spheres	230	(3)
Using vectors to find distances	233	(4)
Point to line; point to plane; line to line; line to plane
Reciprocal vectors	237	(1)
Exercises	238	(6)
Hints and answers	244	(2)
Matrices and vector spaces	246	(76)
Vector spaces	247	(5)
Basis vectors; inner product; some useful inequalities
Linear operators	252	(2)
Matrices	254	(1)
Basic matrix algebra	255	(5)
Matrix addition; multiplication by a scalar; matrix multiplication
Functions of matrices	260	(1)
The transpose of a matrix	260	(1)
The complex and Hermitian conjugates of a matrix	261	(2)
The trace of a matrix	263	(1)
The determinant of a matrix	264	(4)
Properties of determinants
The inverse of a matrix	268	(4)
The rank of a matrix	272	(1)
Special types of square matrix	273	(4)
Diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian and anti-Hermitian; unitary; normal
Eigenvectors and eigenvalues	277	(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	285	(3)
Degenerate eigenvalues
Change of basis and similarity transformations	288	(2)
Diagonalisation of matrices	290	(3)
Quadratic and Hermitian forms	293	(4)
Stationary properties of the eigenvectors; quadratic surfaces
Simultaneous linear equations	297	(15)
Range; null space; N simultaneous linear equations in N unknowns; singular value decomposition
Exercises	312	(7)
Hints and answers	319	(3)
Normal modes	322	(18)
Typical oscillatory systems	323	(5)
Symmetry and normal modes	328	(5)
Rayleigh-Ritz method	333	(2)
Exercises	335	(3)
Hints and answers	338	(2)
Vector calculus	340	(43)
Differentiation of vectors	340	(5)
Composite vector expressions; differential of a vector
Integration of vectors	345	(1)
Space curves	346	(4)
Vector functions of several arguments	350	(1)
Surfaces	351	(2)
Scalar and vector fields	353	(1)
Vector operators	353	(7)
Gradient of a scalar field; divergence of a vector field; curl of a vector field
Vector operator formulae	360	(3)
Vector operators acting on sums and products; combinations of grad, div and curl
Cylindrical and spherical polar coordinates	363	(7)
General curvilinear coordinates	370	(5)
Exercises	375	(6)
Hints and answers	381	(2)
Line, surface and volume integrals	383	(38)
Line integrals	383	(6)
Evaluating line integrals; physical examples; line integrals with respect to a scalar
Connectivity of regions	389	(1)
Green's theorem in a plane	390	(3)
Conservative fields and potentials	393	(2)
Surface integrals	395	(7)
Evaluating surface integrals; vector areas of surfaces; physical examples
Volume integrals	402	(2)
Volumes of three-dimensional regions
Integral forms for grad, div and curl	404	(3)
Divergence theorem and related theorems	407	(5)
Green's theorems; other related integral theorems; physical applications
Stokes' theorem and related theorems	412	(3)
Related integral theorems; physical applications
Exercises	415	(5)
Hints and answers	420	(1)
Fourier series	421	(18)
The Dirichlet conditions	421	(2)
The Fourier coefficients	423	(2)
Symmetry considerations	425	(1)
Discontinuous functions	426	(2)
Non-periodic functions	428	(2)
Integration and differentiation	430	(1)
Complex Fourier series	430	(2)
Parseval's theorem	432	(1)
Exercises	433	(4)
Hints and answers	437	(2)
Integral transforms	439	(35)
Fourier transforms	439	(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	459	(6)
Laplace transforms of derivatives and integrals; other properties of Laplace transforms
Concluding remarks	465	(1)
Exercises	466	(6)
Hints and answers	472	(2)
First-order ordinary differential equations	474	(22)
General form of solution	475	(1)
First-degree first-order equations	476	(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	486	(4)
Equations soluble for p; for x; for y; Clairaut's equation
Exercises	490	(4)
Hints and answers	494	(2)
Higher-order ordinary differential equations	496	(41)
Linear equations with constant coefficients	498	(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	509	(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	524	(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	529	(6)
Hints and answers	535	(2)
Series solutions of ordinary differential equations	537	(44)
Second-order linear ordinary differential equations	537	(4)
Ordinary and singular points
Series solutions about an ordinary point	541	(3)
Series solutions about a regular singular point	544	(5)
Distinct roots not differing by an integer; repeated root of the indicial equation; distinct roots differing by an integer
Obtaining a second solution	549	(5)
The Wronskian method; the derivative method; series form of the second solution
Polynomial solutions	554	(1)
Legendre's equation	555	(9)
General solution for integer l; properties of Legendre polynomials
Bessel's equation	564	(11)
General solution for non-integer v; general solution for integer v; properties of Bessel functions
General remarks	575	(1)
Exercises	575	(4)
Hints and answers	579	(2)
Eigenfunction methods for differential equations	581	(27)
Sets of functions	583	(4)
Some useful inequalities
Adjoint and Hermitian operators	587	(1)
The properties of Hermitian operators	588	(3)
Reality of the eigenvalues; orthogonality of the eigenfunctions; construction of real eigenfunctions
Sturm--Liouville equations	591	(2)
Valid boundary conditions; putting an equation into Sturm--Liouville form
Examples of Sturm--Liouville equations	593	(4)
Legendre's equation; the associated Legendre equation; Bessel's equation; the simple harmonic equation; Hermite's equation; Laguerre's equation; Chebyshev's equation
Superposition of eigenfunctions: Green's functions	597	(4)
A useful generalisation	601	(1)
Exercises	602	(4)
Hints and answers	606	(2)
Partial differential equations: general and particular solutions	608	(38)
Important partial differential equations	609	(4)
The wave equation; the diffusion equation; Laplace's equation; Poisson's equation; Schrodinger's equation
General form of solution	613	(1)
General and particular solutions	614	(12)
First-order equations; inhomogeneous equations and problems; second-order equations
The wave equation	626	(2)
The diffusion equation	628	(4)
Characteristics and the existence of solutions	632	(6)
First-order equations; second-order equations
Uniqueness of solutions	638	(2)
Exercises	640	(4)
Hints and answers	644	(2)
Partial differential equations: separation of variables and other methods	646	(64)
Separation of variables: the general method	646	(4)
Superposition of separated solutions	650	(8)
Separation of variables in polar coordinates	658	(23)
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	681	(5)
Inhomogeneous problems -- Green's functions	686	(16)
Similarities to Green's functions for ordinary differential equations; general boundary-value problems; Dirichlet problems; Neumann problems
Exercises	702	(6)
Hints and answers	708	(2)
Complex variables	710	(66)
Functions of a complex variable	711	(2)
The Cauchy--Riemann relations	713	(3)
Power series in a complex variable	716	(2)
Some elementary functions	718	(3)
Multivalued functions and branch cuts	721	(2)
Singularities and zeroes of complex functions	723	(2)
Complex potentials	725	(5)
Conformal transformations	730	(5)
Applications of conformal transformations	735	(3)
Complex integrals	738	(4)
Cauchy's theorem	742	(3)
Cauchy's integral formula	745	(2)
Taylor and Laurent series	747	(5)
Residue theorem	752	(2)
Location of zeroes	754	(4)
Integrals of sinusoidal functions	758	(1)
Some infinite integrals	759	(3)
Integrals of multivalued functions	762	(2)
Summation of series	764	(1)
Inverse Laplace transform	765	(3)
Exercises	768	(5)
Hints and answers	773	(3)
Tensors	776	(58)
Some notation	777	(1)
Change of basis	778	(1)
Cartesian tensors	779	(2)
First- and zero-order Cartesian tensors	781	(3)
Second- and higher-order Cartesian tensors	784	(3)
The algebra of tensors	787	(1)
The quotient law	788	(2)
The tensors δij and εijk	790	(3)
Isotropic tensors	793	(2)
Improper rotations and pseudotensors	795	(3)
Dual tensors	798	(1)
Physical applications of tensors	799	(4)
Integral theorems for tensors	803	(1)
Non-Cartesian coordinates	804	(2)
The metric tensor	806	(3)
General coordinate transformations and tensors	809	(3)
Relative tensors	812	(2)
Derivatives of basis vectors and Christoffel symbols	814	(3)
Covariant differentiation	817	(3)
Vector operators in tensor form	820	(4)
Absolute derivatives along curves	824	(1)
Geodesics	825	(1)
Exercises	826	(5)
Hints and answers	831	(3)
Calculus of variations	834	(28)
The Euler--Lagrange equation	835	(1)
Special cases	836	(4)
F does not contain y explicitly; F does not contain x explicitly
Some extensions	840	(4)
Several dependent variables; several independent variables; higher-order derivatives; variable end-points
Constrained variation	844	(2)
Physical variational principles	846	(3)
Fermat's principle in optics; Hamilton's principle in mechanics
General eigenvalue problems	849	(2)
Estimation of eigenvalues and eigenfunctions	851	(3)
Adjustment of parameters	854	(2)
Exercises	856	(4)
Hints and answers	860	(2)
Integral equations	862	(21)
Obtaining an integral equation from a differential equation	862	(1)
Types of integral equation	863	(1)
Operator notation and the existence of solutions	864	(1)
Closed-form solutions	865	(7)
Separable kernels; integral transform methods; differentiation
Neumann series	872	(2)
Fredholm theory	874	(1)
Schmidt--Hilbert theory	875	(3)
Exercises	878	(4)
Hints and answers	882	(1)
Group theory	883	(35)
Groups	883	(8)
Definition of a group; examples of groups
Finite groups	891	(3)
Non-Abelian groups	894	(4)
Permutation groups	898	(3)
Mappings between groups	901	(2)
Subgroups	903	(2)
Subdividing a group	905	(7)
Equivalence relations and classes; congruence and cosets; conjugates and classes
Exercises	912	(3)
Hints and answers	915	(3)
Representation theory	918	(43)
Dipole moments of molecules	919	(1)
Choosing an appropriate formalism	920	(6)
Equivalent representations	926	(2)
Reducibility of a representation	928	(4)
The orthogonality theorem for irreducible representations	932	(2)
Characters	934	(3)
Orthogonality property of characters
Counting irreps using characters	937	(5)
Summation rules for irreps
Construction of a character table	942	(2)
Group nomenclature	944	(1)
Product representations	945	(2)
Physical applications of group theory	947	(8)
Bonding in molecules; matrix elements in quantum mechanics; degeneracy of normal modes; breaking of degeneracies
Exercises	955	(4)
Hints and answers	959	(2)
Probability	961	(103)
Venn diagrams	961	(5)
Probability	966	(9)
Axioms and theorems; conditional probability; Bayes' theorem
Permutations and combinations	975	(6)
Random variables and distributions	981	(4)
Discrete random variables; continuous random variables
Properties of distributions	985	(7)
Mean; mode and median; variance and standard deviation; moments; central moments
Functions of random variables	992	(7)
Generating functions	999	(10)
Probability generating functions; moment generating functions; characteristic functions; cumulant generating functions
Important discrete distributions	1009	(12)
Binomial; geometric; negative binomial; hypergeometric; Poisson
Important continuous distributions	1021	(15)
Gaussian; log-normal; exponential; gamma; chi-squared; Cauchy; breit--Wigner; uniform
The central limit theorem	1036	(2)
Joint distributions	1038	(3)
Discrete bivariate; continuous bivariate; marginal and conditional distributions
Properties of joint distributions	1041	(6)
Means; variances; covariance and correlation
Generating functions for joint distributions	1047	(1)
Transformation of variables in joint distributions	1048	(1)
Important joint distributions	1049	(4)
Multinominal; multivariate Guassian
Exercises	1053	(8)
Hints and answers	1061	(3)
Statistics	1064	(84)
Experiments, samples and populations	1064	(1)
Sample statistics	1065	(7)
Averages; variance and standard deviation; moments; covariance and correlation
Estimators and sampling distributions	1072	(14)
Consistency, bias and efficiency; Fisher's inequality; standard errors; confidence limits
Some basic estimators	1086	(11)
Mean; variance; standard deviation; moments; covariance and correlation
Maximum-likelihood method	1097	(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	1113	(6)
Linear least squares; non-linear least squares
Hypothesis testing	1119	(21)
Simple and composite hypotheses; statistical tests; Neyman--Pearson; generalised likelihood-ratio; Student's t; Fisher's F; goodness of fit
Exercises	1140	(5)
Hints and answers	1145	(3)
Numerical methods	1148	(53)
Algebraic and transcendental equations	1149	(7)
Rearrangement of the equation; linear interpolation; binary chopping; Newton--Raphson method
Convergence of iteration schemes	1156	(2)
Simultaneous linear equations	1158	(6)
Gaussian elimination; Gauss-Seidel iteration; tridiagonal matrices
Numerical integration	1164	(15)
Trapezium rule; Simpson's rule; Gaussian integration; Monte Carlo methods
Finite differences	1179	(1)
Differential equations	1180	(8)
Difference equations; Taylor series solutions; prediction and correction; Runge--Kutta methods; isoclines
Higher-order equations	1188	(2)
Partial differential equations	1190	(3)
Exercises	1193	(5)
Hints and answers	1198	(3)
Appendix Gamma, beta and error functions	1201	(5)
A.1.1 The gamma function	1201	(2)
A.1.2 The beta function	1203	(1)
A.1.3 The error function	1204	(2)
Index	1206