com Mathematics for Physicists www.com The Manchester Physics Series General Editors J. LOEBINGER School of Physics and Astronomy, University of Manchester Properties of Matter B. Mendoza Statistical Physics F. Mandl Second Edition Electromagnetism l.
Phillips Second Edition Statistics R. Barlow Solid State Physics J. Hall Second Edition Quantum Mechanics F. Mandl Computing for Scientists R.
Barnett The Physics of Stars A. Phillips Second Edition Nuclear Physics J. Lilley Introduction to Quantum Mechanics A. Phillips Particle Physics B.
Shaw Third Edition Dynamics and Relativity J. Smith Vibrations and Waves G. King Mathematics for Physicists B.com Mathematics for Physicists B. MARTIN Department of Physics and Astronomy University College London G.
SHAW Department of Physics and Astronomy Manchester University www.com This edition first published 2015 © 2015 John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.
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pages cm Includes bibliographical references and index. ISBN 978-0-470-66023-2 (cloth) – ISBN 978-0-470-66022-5 (pbk.M35 2015 510–dc23 2015008518 Set in 11/13pt Computer Modern by Aptara Inc., New Delhi, India.com Contents Editors’ preface to the Manchester Physics Series xi Authors’ preface xiii Notes and website information xv 1 Real numbers, variables and functions 1 1.1 Rules of arithmetic: rational and irrational numbers 1 1.2 Factors, powers and rationalisation 4 *1.1 Rules of elementary algebra √ 9 *1.2 Proof of the irrationality of 2 11 1.3 Formulas, identities and equations 11 1.4 The binomial theorem 13 1.5 Absolute values and inequalities 17 1.3 Functions, graphs and co-ordinates 20 1.2 Cartesian co-ordinates 23 Problems 1 28 2 Some basic functions and equations 31 2.2 Rational functions and partial fractions 37 2.3 Algebraic and transcendental functions 41 2.1 Angles and polar co-ordinates 41 2.2 Sine and cosine 44 2.3 More trigonometric functions 46 2.4 Trigonometric identities and equations 48 2.5 Sine and cosine rules 51 2.3 Logarithms and exponentials 53 2.1 The laws of logarithms 54 2.4 Conic sections 63 Problems 2 68 www.com vi Contents 3 Differential calculus 71 3.1 Limits and continuity 71 3.2 Some standard derivatives 80 3.5 More standard derivatives 87 3.4 Higher derivatives and stationary points 90 3.5 Curve sketching 95 Problems 3 98 4 Integral calculus 101 4.1 Indefinite integrals 101 4.2 Definite integrals 104 4.1 Integrals and areas 105 4.3 Change of variables and substitutions 111 4.1 Change of variables 111 4.2 Products of sines and cosines 113 4.5 More standard integrals 117 4.7 Symmetric and antisymmetric integrals 119 4.4 Integration by parts 120 4.1 Infinite integrals 126 4.7 Applications of integration 132 4.1 Work done by a varying force 132 4.2 The length of a curve 133 *4.3 Surfaces and volumes of revolution 134 *4.4 Moments of inertia 136 Problems 4 137 5 Series and expansions 143 5.2 Convergence of infinite series 146 www.com Contents vii 5.3 Taylor’s theorem and its applications 149 5.2 Small changes and l’Hôpital’s rule 150 5.4 Approximation errors: Euler’s number 153 5.1 Taylor and Maclaurin series 154 5.2 Operations with series 157 *5.5 Proof of d’Alembert’s ratio test 161 *5.6 Alternating and other series 163 Problems 5 165 6 Complex numbers and variables 169 6.2 Complex plane: Argand diagrams 172 6.3 Complex variables and series 176 *6.1 Proof of the ratio test for complex series 179 6.1 Powers and roots 182 6.2 Exponentials and logarithms 184 6.3 De Moivre’s theorem 185 *6.4 Summation of series and evaluation of integrals 187 Problems 6 189 7 Partial differentiation 191 7.1 Two standard results 195 7.2 Exact differentials 197 7.3 The chain rule 198 7.4 Homogeneous functions and Euler’s theorem 199 7.3 Change of variables 200 7.7 Differentiation of integrals 211 Problems 7 214 8 Vectors 219 8.1 Scalars and vectors 219 8.2 Components of vectors: Cartesian co-ordinates 221 8.2 Products of vectors 225 8.2 Vector product 228 www.com viii Contents 8.3 Applications to geometry 238 8.4 Differentiation and integration 243 Problems 8 246 9 Determinants, Vectors and Matrices 249 9.1 General properties of determinants 253 9.2 Homogeneous linear equations 257 9.2 Vectors in n Dimensions 260 9.3 Matrices and linear transformations 265 9.3 Transpose, complex, and Hermitian conjugates 273 9.1 Some special square matrices 274 9.2 The determinant of a matrix 276 9.4 Inhomogeneous simultaneous linear equations 282 Problems 9 284 10 Eigenvalues and eigenvectors 291 10.1 The eigenvalue equation 291 10.1 Properties of eigenvalues 293 10.2 Properties of eigenvectors 296 10.2 Diagonalisation of matrices 302 *10.1 Normal modes of oscillation 305 *10.2 Quadratic forms 308 Problems 10 312 11 Line and multiple integrals 315 11.1 Line integrals in a plane 315 11.2 Integrals around closed contours and along arcs 319 11.3 Line integrals in three dimensions 321 11.1 Green’s theorem in the plane and perfect differentials 326 11.2 Other co-ordinate systems and change of variables 330 11.3 Curvilinear co-ordinates in three dimensions 333 11.1 Cylindrical and spherical polar co-ordinates 334 www.com Contents ix 11.4 Triple or volume integrals 337 11.1 Change of variables 338 Problems 11 340 12 Vector calculus 345 12.1 Scalar and vector fields 345 12.1 Gradient of a scalar field 346 12.2 Div, grad and curl 349 12.3 Orthogonal curvilinear co-ordinates 352 12.2 Line, surface, and volume integrals 355 12.2 Conservative fields and potentials 359 12.4 Volume integrals: moments of inertia 367 12.3 The divergence theorem 368 12.1 Proof of the divergence theorem and Green’s identities 369 *12.2 Divergence in orthogonal curvilinear co-ordinates 372 *12.3 Poisson’s equation and Gauss’ theorem 373 *12.4 The continuity equation 376 12.1 Proof of Stokes’ theorem 378 *12.2 Curl in curvilinear co-ordinates 380 *12.3 Applications to electromagnetic fields 381 Problems 12 384 13 Fourier analysis 389 13.1 Fourier coefficients 390 13.3 Change of period 398 13.4 Non-periodic functions 399 13.5 Integration and differentiation of Fourier series 401 13.6 Mean values and Parseval’s theorem 405 13.2 Complex Fourier series 407 *13.1 Fourier expansions and vector spaces 409 13.1 Properties of Fourier transforms 414 *13.2 The Dirac delta function 419 *13.3 The convolution theorem 423 Problems 13 426 14 Ordinary differential equations 431 14.1 First-order equations 433 14.2 Separation of variables 434 14.3 Homogeneous equations 435 www.5 First-order linear equations 440 14.2 Linear ODEs with constant coefficients 441 14.2 Particular integrals: method of undetermined coefficients 446 *14.3 Particular integrals: the D-operator method 448 *14.3 Euler’s equation 459 Problems 14 461 15 Series solutions of ordinary differential equations 465 15.1 Series solutions about a regular point 467 15.2 Series solutions about a regular singularity: Frobenius method 469 15.1 Legendre functions and Legendre polynomials 482 *15.2 The generating function 487 *15.3 Associated Legendre equation 490 *15.2 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 16.1 The wave equation in one spatial dimension 512 16.2 The wave equation in three spatial dimensions 515 16.3 The diffusion equation in one spatial dimension 518 16.3 Separation of variables: polar co-ordinates 520 16.1 Plane-polar co-ordinates 520 16.2 Spherical polar co-ordinates 524 16.3 Cylindrical polar co-ordinates 529 *16.4 The wave equation: d’Alembert’s solution 532 *16.6 Boundary conditions and uniqueness 538 *16.1 Laplace transforms 540 Problems 16 544 Answers to selected problems 549 Index 559 www.com Editors’ preface to the Manchester Physics Series The Manchester Physics Series is a set of textbooks at first degree level. It grew out of the experience at the University of Manchester, widely shared elsewhere, that many textbooks con- tain much more material than can be accommodated in a typical undergraduate course; and that this material is only rarely so arranged as to allow the definition of a short self-contained course. The plan for this series was to produce short books so that lecturers would find them attractive for undergraduate courses, and so that students would not be frightened off by their encyclopaedic size or price.
To achieve this, we have been very selective in the choice of topics, with the emphasis on the basic physics together with some instructive, stimulating and useful applications. Although these books were conceived as a series, each of them is self-contained and can be used independently of the others. Several of them are suitable for wider use in other sciences. Each Author’s Preface gives details about the level, prerequisites, etc., of that volume.
The Manchester Physics Series has been very successful since its inception over 40 years ago, with total sales of more than a quarter of a million copies. We are extremely grateful to the many students and colleagues, at Manchester and elsewhere, for helpful criticisms and stimulating comments. Our particular thanks go to the authors for all the work they have done, for the many new ideas they have contributed, and for discussing patiently, and often accepting, the suggestions of the editors. Finally, we would like to thank our publisher, John Wiley & Sons, Ltd., for their enthusiastic and continued commitment to the Manchester Physics Series.
Loebinger August 2014 www.com Authors’ preface Our aim in writing this book is to produce a relatively short volume that covers all the essential mathematics needed for a typical first degree in physics, from a starting point that is compatible with modern school mathematics syllabuses.