Mathematical Methods for Physics and Engineering: A Comprehensive Guide

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

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ISBN: 9780521679718 | 0521679710
Cover: Paperback
Copyright: 3/13/2006

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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.

Preface to the third edition

Preface to the second edition

xxiii

Preface to the first edition

xxv

Preliminary algebra

(40)

Simple functions and equations

Polynomial equations: factorisation; properties of roots

(9)

Trigonometric identities

Single angle; compound angles; double- and half-angle identities

(5)

Coordinate geometry

(3)

Partial fractions

Complications and special cases

(7)

Binomial expansion

(2)

Properties of binomial coefficients

(3)

Some particular methods of proof

Proof by induction; proof by contradiction; necessary and sufficient conditions

(6)

Exercises

(3)

Hints and answers

(2)

Preliminary calculus

(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

(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

(17)

Exercises

(5)

Hints and answers

(2)

Complex numbers and hyperbolic functions

(32)

The need for complex numbers

(2)

Manipulation of complex numbers

Addition and subtraction; modulus and argument; multiplication; complex conjugate; division

(7)

Polar representation of complex numbers

Multiplication and division in polar form

(3)

de Moivre's theorem

trigonometric identities; finding the nth roots of unity; solving polynomial equations

(4)

Complex logarithms and complex powers

(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

(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

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Mathematical Methods for Physics and Engineering: A Comprehensive Guide

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Table of Contents

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