Mathematical Methods for Physicists: A Concise Introduction

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ISBN: 9780521652278 | 0521652278
Cover: Hardcover
Copyright: 7/31/2000

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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.

Preface

Vector and tensor analysis

(61)

Vectors and scalars

(2)

Direction angles and direction cosines

(1)

Vector algebra

(1)

Equality of vectors

(1)

Vector addition

(1)

Multiplication by a scalar

(1)

The scalar product

(2)

The vector (cross or outer) product

(3)

The triple scalar product A . (B x C)

(1)

The triple vector product

(1)

Change of coordinate system

(2)

The linear vector space Vn

(2)

Vector differentiation

(1)

Space curves

(1)

Motion in a plane

(1)

A vector treatment of classical orbit theory

(2)

Vector differential of a scalar field and the gradient

(1)

Conservative vector field

(1)

The vector differential operator ▾

(1)

Vector differentiation of a vector field

(5)

The divergence of a vector

(2)

The operator ▾2, the Laplacian

(1)

The curl of a vector

(3)

Formulas involving ▾

(1)

Orthogonal curvilinear coordinates

(5)

Special orthogonal coordinate systems

(3)

Cylindrical coordinates (ρ, &phis;, z)

(2)

Spherical coordinates (r, &thetas;, &phis;)

(1)

Vector integration and integral theorems

(9)

Gauss' theorem (the divergence theorem)

(2)

Continuity equation

(1)

Stokes' theorem

(3)

Green's theorem

(1)

Green's theorem in the plane

(1)

Helmholtz's theorem

(1)

Some useful integral relations

(2)

Tensor analysis

(1)

Contravariant and covariant vectors

(1)

Tensors of second rank

(1)

Basic operations with tensors

(1)

Quotient law

(1)

The line element and metric tensor

(2)

Associated tensors

(1)

Geodesics in a Riemannian space

(2)

Covariant differentiation

(2)

Problems

(5)

Ordinary differential equations

(38)

First-order differential equations

(9)

Separable variables

(4)

Exact equations

(2)

Integrating factors

(3)

Bernoulli's equation

(1)

Second-order equations with constant coefficients

(11)

Nature of the solution of linear equations

(1)

General solutions of the second-order equations

(1)

Finding the complementary function

(3)

Finding the particular integral

(1)

Particular integral and the operator D(= d / dx)

(1)

Rules for D operators

(4)

The Euler linear equation

(2)

Solutions in power series

(8)

Ordinary and singular points of a differential equation

(1)

Frobenius and Fuchs theorem

(7)

Simultaneous equations

(1)

The gamma and beta functions

(2)

Problems

(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)

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Mathematical Methods for Physicists: A Concise Introduction

Summary

Table of Contents

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