Brian Martin was a full-time member of staff of the Department of Physics & Astronomy at UCL from 1968 to 1995, including a decade from 1994 to 2004 as Head of the Department. I retired in 2005 and now hold the title of Emeritus Professor of Physics. I have extensive experience of teaching undergraduate mathematics classes at all levels and experience of other universities via external examining for first degrees at Imperial College and Royal Holloway College London. I was also the external member of the General Board of the Department of Physics at Cambridge University that reviewed the whole academic programme of that department, including teaching.

Graham Shaw is a full-time member of staff of the School of Physics & Astronomy at Manchester University and will retire in September 2009. I have extensive experience of teaching undergraduate physics and the associated mathematics, and have been a member of the department’s Teaching Committee and the Course Director of the Honours School of Mathematics and Physics for many years.

Authors’ preface xiii

Notes and website information xv

1 Real numbers, variables and functions 1

1.1 Real numbers 1

1.1.1 Rules of arithmetic: rational and irrational numbers 1

1.1.2 Factors, powers and rationalisation 4

1.1.3 Number systems 6

1.2 Real variables 9

1.2.1 Rules of elementary algebra 9

1.2.2 Proof of the irrationality of 2 11

1.2.3 Formulas, identities and equations 11

1.2.4 The binomial theorem 13

1.2.5 Absolute values and inequalities 17

1.3 Functions, graphs and co-ordinates 20

1.3.1 Functions 20

1.3.2 Cartesian co-ordinates 23

Problems 1 28

2 Some basic functions and equations 31

2.1 Algebraic functions 31

2.1.1 Polynomials 31

2.1.2 Rational functions and partial fractions 37

2.1.3 Algebraic and transcendental functions 41

2.2 Trigonometric functions 41

2.2.1 Angles and polar co-ordinates 41

2.2.2 Sine and cosine 44

2.2.3 More trigonometric functions 46

2.2.4 Trigonometric identities and equations 48

2.2.5 Sine and cosine rules 51

2.3 Logarithms and exponentials 53

2.3.1 The laws of logarithms 54

2.3.2 Exponential function 56

2.3.3 Hyperbolic functions 60

2.4 Conic sections 63

Problems 2 68

3 Differential calculus 71

3.1 Limits and continuity 71

3.1.1 Limits 71

3.1.2 Continuity 75

3.2 Differentiation 77

3.2.1 Differentiability 78

3.2.2 Some standard derivatives 80

3.3 General methods 82

3.3.1 Product rule 83

3.3.2 Quotient rule 83

3.3.3 Reciprocal relation 84

3.3.4 Chain rule 86

3.3.5 More standard derivatives 87

3.3.6 Implicit functions 89

3.4 Higher derivatives and stationary points 90

3.4.1 Stationary points 92

3.5 Curve sketching 95

Problems 3 98

4 Integral calculus 101

4.1 Indefinite integrals 101

4.2 Definite integrals 104

4.2.1 Integrals and areas 105

4.2.2 Riemann integration 108

4.3 Change of variables and substitutions 111

4.3.1 Change of variables 111

4.3.2 Products of sines and cosines 113

4.3.3 Logarithmic integration 115

4.3.4 Partial fractions 116

4.3.5 More standard integrals 117

4.3.6 Tangent substitutions 118

4.3.7 Symmetric and antisymmetric integrals 119

4.4 Integration by parts 120

4.5 Numerical integration 123

4.6 Improper integrals 126

4.6.1 Infinite integrals 126

4.6.2 Singular integrals 129

4.7 Applications of integration 132

4.7.1 Work done by a varying force 132

4.7.2 The length of a curve 133

4.7.3 Surfaces and volumes of revolution 134

4.7.4 Moments of inertia 136

Problems 4 137

5 Series and expansions 143

5.1 Series 143

5.2 Convergence of infinite series 146

5.3 Taylor’s theorem and its applications 149

5.3.1 Taylor’s theorem 149

5.3.2 Small changes and l’Hˆopital’s rule 150

5.3.3 Newton’s method 152

5.3.4 Approximation errors: Euler’s number 153

5.4 Series expansions 153

5.4.1 Taylor and Maclaurin series 154

5.4.2 Operations with series 157

5.5 Proof of d’Alembert’s ratio test 161

5.5.1 Positive series 161

5.5.2 General series 162

5.6 Alternating and other series 163

Problems 5 165

6 Complex numbers and variables 169

6.1 Complex numbers 169

6.2 Complex plane: Argand diagrams 172

6.3 Complex variables and series 176

6.3.1 Proof of the ratio test for complex series 179

6.4 Euler’s formula 180

6.4.1 Powers and roots 182

6.4.2 Exponentials and logarithms 184

6.4.3 De Moivre’s theorem 185

6.4.4 Summation of series and evaluation of integrals 187

Problems 6 189

7 Partial differentiation 191

7.1 Partial derivatives 191

7.2 Differentials 193

7.2.1 Two standard results 195

7.2.2 Exact differentials 197

7.2.3 The chain rule 198

7.2.4 Homogeneous functions and Euler’s theorem 199

7.3 Change of variables 200

7.4 Taylor series 203

7.5 Stationary points 206

*7.6 Lagrange multipliers 209

7.7 Differentiation of integrals 211

Problems 7 214

8 Vectors 219

8.1 Scalars and vectors 219

8.1.1 Vector algebra 220

8.1.2 Components of vectors: Cartesian co-ordinates 221

8.2 Products of vectors 225

8.2.1 Scalar product 225

8.2.2 Vector product 228

8.2.3 Triple products 231

8.2.4 Reciprocal vectors 236

8.3 Applications to geometry 238

8.3.1 Straight lines 238

8.3.2 Planes 241

8.4 Differentiation and integration 243

Problems 8 246

9 Determinants, Vectors and Matrices 249

9.1 Determinants 249

9.1.1 General properties of determinants 253

9.1.2 Homogeneous linear equations 257

9.2 Vectors in n Dimensions 260

9.2.1 Basis vectors 261

9.2.2 Scalar products 263

9.3 Matrices and linear transformations 265

9.3.1 Matrices 265

9.3.2 Linear transformations 270

9.3.3 Transpose, complex, and Hermitian conjugates 273

9.4 Square Matrices 274

9.4.1 Some special square matrices 274

9.4.2 The determinant of a matrix 276

9.4.3 Matrix inversion 278

9.4.4 Inhomogeneous simultaneous linear equations 282

Problems 9 284

10 Eigenvalues and eigenvectors 291

10.1 The eigenvalue equation 291

10.1.1 Properties of eigenvalues 293

10.1.2 Properties of eigenvectors 296

10.1.3 Hermitian matrices 299

10.2 Diagonalisation of matrices 302

10.2.1 Normal modes of oscillation 305

10.2.2 Quadratic forms 308

Problems 10 312

11 Line and multiple integrals 315

11.1 Line integrals 315

11.1.1 Line integrals in a plane 315

11.1.2 Integrals around closed contours and along arcs 319

11.1.3 Line integrals in three dimensions 321

11.2 Double integrals 323

11.2.1 Green’s theorem in the plane and perfect differentials 326

11.2.2 Other co-ordinate systems and change of variables 330

11.3 Curvilinear co-ordinates in three dimensions 333

11.3.1 Cylindrical and spherical polar co-ordinates 334

11.4 Triple or volume integrals 337

11.4.1 Change of variables 338

Problems 11 340

12 Vector calculus 345

12.1 Scalar and vector fields 345

12.1.1 Gradient of a scalar field 346

12.1.2 Div, grad and curl 349

12.1.3 Orthogonal curvilinear co-ordinates 352

12.2 Line, surface, and volume integrals 355

12.2.1 Line integrals 355

12.2.2 Conservative fields and potentials 359

12.2.3 Surface integrals 362

12.2.4 Volume integrals: moments of inertia 367

12.3 The divergence theorem 368

12.3.1 Proof of the divergence theorem and Green’s identities 369

12.3.2 Divergence in orthogonal curvilinear co-ordinates 372

12.3.3 Poisson’s equation and Gauss’ theorem 373

12.3.4 The continuity equation 376

12.4 Stokes’ theorem 377

12.4.1 Proof of Stokes’ theorem 378

12.4.2 Curl in curvilinear co-ordinates 380

12.4.3 Applications to electromagnetic fields 381

Problems 12 384

13 Fourier analysis 389

13.1 Fourier series 389

13.1.1 Fourier coefficients 390

13.1.2 Convergence 394

13.1.3 Change of period 398

13.1.4 Non-periodic functions 399

13.1.5 Integration and differentiation of Fourier series 401

13.1.6 Mean values and Parseval’s theorem 405

13.2 Complex Fourier series 407

13.2.1 Fourier expansions and vector spaces 409

13.3 Fourier transforms 410

13.3.1 Properties of Fourier transforms 414

13.3.2 The Dirac delta function 419

13.3.3 The convolution theorem 423

Problems 13 426

14 Ordinary differential equations 431

14.1 First-order equations 433

14.1.1 Direct integration 433

14.1.2 Separation of variables 434

14.1.3 Homogeneous equations 435

14.1.4 Exact equations 438

14.1.5 First-order linear equations 440

14.2 Linear ODEs with constant coefficients 441

14.2.1 Complementary functions 442

14.2.2 Particular integrals: method of undetermined coefficients 446

14.2.3 Particular integrals: the D-operator method 448

14.2.4 Laplace transforms 453

14.3 Euler’s equation 459

Problems 14 461

15 Series solutions of ordinary differential equations 465

15.1 Series solutions 465

15.1.1 Series solutions about a regular point 467

15.1.2 Series solutions about a regular singularity: Frobenius method 469

15.1.3 Polynomial solutions 475

15.2 Eigenvalue equations 478

15.3 Legendre’s equation 481

15.3.1 Legendre functions and Legendre polynomials 482

15.3.2 The generating function 487

15.3.3 Associated Legendre equation 490

15.3.4 Rodrigues’ formula 492

15.4 Bessel’s equation 494

15.4.1 Bessel functions 495

15.4.2 Properties of non-singular Bessel functions Jν (x) 499

Problems 15 502

16 Partial differential equations 507

16.1 Some important PDEs in physics 510

16.2 Separation of variables: Cartesian co-ordinates 511

16.2.1 The wave equation in one spatial dimension 512

16.2.2 The wave equation in three spatial dimensions 515

16.2.3 The diffusion equation in one spatial dimension 518

16.3 Separation of variables: polar co-ordinates 520

16.3.1 Plane-polar co-ordinates 520

16.3.2 Spherical polar co-ordinates 524

16.3.3 Cylindrical polar co-ordinates 529

16.4 The wave equation: d’Alembert’s solution 532

16.5 Euler equations 535

16.6 Boundary conditions and uniqueness 538

16.6.1 Laplace transforms 540

Problems 16 544

Answers to selected problems 549

Index 559

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