Mathematical Methods For Physics And Engineering Hobson Pdf

Unformatted text preview: This page intentionally left blank Mathematical Methods for Physics and Engineering The third edition of this highly acclaimed undergraduate textbook is suitable for teaching all the mathematics ever likely to be needed for an undergraduate course in any of the physical sciences. As well as lucid descriptions of all the topics covered and many worked examples, it contains more than 800 exercises. A number of additional topics have been included and the text has undergone signiﬁcant reorganisation in some areas. 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 including WKB methods and saddle-point integration techniques • provide an introduction to quantum operators. Further tabulations, of relevance in statistics and numerical integration, have been added. In this edition, all 400 odd-numbered exercises are provided with complete worked solutions in a separate manual, available to both students and their teachers; these are in addition to the hints and outline answers given in the main text. The even-numbered exercises have no hints, answers or worked solutions and can be used for unaided homework; full solutions to them are available to instructors on a password-protected website. K e n R i l e y read mathematics at the University of Cambridge and proceeded to a Ph.D. there in theoretical and experimental nuclear physics. He became a research associate in elementary particle physics at Brookhaven, and then, having taken up a lectureship at the Cavendish Laboratory, Cambridge, continued this research at the Rutherford Laboratory and Stanford; in particular he was involved in the experimental discovery of a number of the early baryonic resonances. As well as having been Senior Tutor at Clare College, where he has taught physics and mathematics for over 40 years, he has served on many committees concerned with the teaching and examining of these subjects at all levels of tertiary and undergraduate education. He is also one of the authors of 200 Puzzling Physics Problems. M i c h a e l H o b s o n read natural sciences at the University of Cambridge, specialising in theoretical physics, and remained at the Cavendish Laboratory to complete a Ph.D. in the physics of star-formation. As a research fellow at Trinity Hall, Cambridge and subsequently an advanced fellow of the Particle Physics and Astronomy Research Council, he developed an interest in cosmology, and in particular in the study of ﬂuctuations in the cosmic microwave background. He was involved in the ﬁrst detection of these ﬂuctuations using a ground-based interferometer. He is currently a University Reader at the Cavendish Laboratory, his research interests include both theoretical and observational aspects of cosmology, and he is the principal author of General Relativity: An Introduction for Physicists. He is also a Director of Studies in Natural Sciences at Trinity Hall and enjoys an active role in the teaching of undergraduate physics and mathematics. S t e p h e n B e n c e obtained both his undergraduate degree in Natural Sciences and his Ph.D. in Astrophysics from the University of Cambridge. He then became a Research Associate with a special interest in star-formation processes and the structure of star-forming regions. In particular, his research concentrated on the physics of jets and outﬂows from young stars. He has had considerable experience of teaching mathematics and physics to undergraduate and pre-universtiy students. ii Mathematical Methods for Physics and Engineering Third Edition K. F. RILEY, M. P. HOBSON and S. J. BENCE CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York Information on this title: © K. F. Riley, M. P. Hobson and S. J. Bence 2006 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2006 ISBN-13 978-0-511-56904-3 eBook (Dawsonera) ISBN-13 978-0-521-86153-3 hardback ISBN-13 978-0-521-67971-8 paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface to the third edition Preface to the second edition Preface to the ﬁrst edition 1 1.1 page xx xxiii xxv Preliminary algebra Simple functions and equations 1 1 Polynomial equations; factorisation; properties of roots 1.2 Trigonometric identities 10 Single angle; compound angles; double- and half-angle identities 1.3 1.4 Coordinate geometry Partial fractions 15 18 Complications and special cases 1.5 1.6 1.7 Binomial expansion Properties of binomial coeﬃcients Some particular methods of proof 25 27 30 Proof by induction; proof by contradiction; necessary and suﬃcient conditions 1.8 1.9 Exercises Hints and answers 36 39 2 2.1 Preliminary calculus Diﬀerentiation 41 41 Diﬀerentiation from ﬁrst principles; products; the chain rule; quotients; implicit diﬀerentiation; logarithmic diﬀerentiation; Leibnitz' theorem; special points of a function; curvature; theorems of diﬀerentiation v CONTENTS 2.2 Integration 59 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 2.3 2.4 Exercises Hints and answers 76 81 3 3.1 3.2 Complex numbers and hyperbolic functions The need for complex numbers Manipulation of complex numbers 83 83 85 Addition and subtraction; modulus and argument; multiplication; complex conjugate; division 3.3 Polar representation of complex numbers 92 Multiplication and division in polar form 3.4 de Moivre's theorem 95 trigonometric identities; ﬁnding the nth roots of unity; solving polynomial equations 3.5 3.6 3.7 Complex logarithms and complex powers Applications to diﬀerentiation and integration Hyperbolic functions 99 101 102 Deﬁnitions; hyperbolic–trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculus of hyperbolic functions 3.8 3.9 Exercises Hints and answers 109 113 4 4.1 4.2 Series and limits Series Summation of series 115 115 116 Arithmetic series; geometric series; arithmetico-geometric series; the diﬀerence method; series involving natural numbers; transformation of series 4.3 Convergence of inﬁnite series 124 Absolute and conditional convergence; series containing only real positive terms; alternating series test 4.4 4.5 Operations with series Power series 131 131 Convergence of power series; operations with power series 4.6 Taylor series 136 Taylor's theorem; approximation errors; standard Maclaurin series 4.7 4.8 4.9 Evaluation of limits Exercises Hints and answers 141 144 149 vi CONTENTS 5 Partial diﬀerentiation 151 5.1 Deﬁnition of the partial derivative 151 5.2 The total diﬀerential and total derivative 153 5.3 Exact and inexact diﬀerentials 155 5.4 Useful theorems of partial diﬀerentiation 157 5.5 The chain rule 157 5.6 Change of variables 158 5.7 Taylor's theorem for many-variable functions 160 5.8 Stationary values of many-variable functions 162 5.9 Stationary values under constraints 167 5.10 Envelopes 173 5.11 Thermodynamic relations 176 5.12 Diﬀerentiation of integrals 178 5.13 Exercises 179 5.14 Hints and answers 185 6 Multiple integrals 187 6.1 Double integrals 187 6.2 Triple integrals 190 6.3 Applications of multiple integrals 191 Areas and volumes; masses, centres of mass and centroids; Pappus' theorems; moments of inertia; mean values of functions 6.4 Change of variables in multiple integrals 199 Change ∞ −x2 of variables in double integrals; evaluation of the integral I = e dx; change of variables in triple integrals; general properties of −∞ Jacobians 6.5 Exercises 207 6.6 Hints and answers 211 7 Vector algebra 212 7.1 Scalars and vectors 212 7.2 Addition and subtraction of vectors 213 7.3 Multiplication by a scalar 214 7.4 Basis vectors and components 217 7.5 Magnitude of a vector 218 7.6 Multiplication of vectors 219 Scalar product; vector product; scalar triple product; vector triple product vii CONTENTS 7.7 7.8 Equations of lines, planes and spheres Using vectors to ﬁnd distances 226 229 Point to line; point to plane; line to line; line to plane 7.9 7.10 7.11 Reciprocal vectors Exercises Hints and answers 233 234 240 8 8.1 Matrices and vector spaces Vector spaces 241 242 Basis vectors; inner product; some useful inequalities 8.2 8.3 8.4 Linear operators Matrices Basic matrix algebra 247 249 250 Matrix addition; multiplication by a scalar; matrix multiplication 8.5 8.6 8.7 8.8 8.9 Functions of matrices The transpose of a matrix The complex and Hermitian conjugates of a matrix The trace of a matrix The determinant of a matrix 255 255 256 258 259 Properties of determinants 8.10 8.11 8.12 The inverse of a matrix The rank of a matrix Special types of square matrix 263 267 268 Diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian and anti-Hermitian; unitary; normal 8.13 Eigenvectors and eigenvalues 272 Of a normal matrix; of Hermitian and anti-Hermitian matrices; of a unitary matrix; of a general square matrix 8.14 Determination of eigenvalues and eigenvectors 280 Degenerate eigenvalues 8.15 8.16 8.17 Change of basis and similarity transformations Diagonalisation of matrices Quadratic and Hermitian forms 282 285 288 Stationary properties of the eigenvectors; quadratic surfaces 8.18 Simultaneous linear equations 292 Range; null space; N simultaneous linear equations in N unknowns; singular value decomposition 8.19 8.20 Exercises Hints and answers 307 314 9 9.1 9.2 Normal modes Typical oscillatory systems Symmetry and normal modes 316 317 322 viii CONTENTS 9.3 9.4 9.5 Rayleigh–Ritz method Exercises Hints and answers 327 329 332 10 10.1 Vector calculus Diﬀerentiation of vectors 334 334 Composite vector expressions; diﬀerential of a vector 10.2 10.3 10.4 10.5 10.6 10.7 Integration of vectors Space curves Vector functions of several arguments Surfaces Scalar and vector ﬁelds Vector operators 339 340 344 345 347 347 Gradient of a scalar ﬁeld; divergence of a vector ﬁeld; curl of a vector ﬁeld 10.8 Vector operator formulae 354 Vector operators acting on sums and products; combinations of grad, div and curl 10.9 10.10 10.11 10.12 Cylindrical and spherical polar coordinates General curvilinear coordinates Exercises Hints and answers 357 364 369 375 11 11.1 Line, surface and volume integrals Line integrals 377 377 Evaluating line integrals; physical examples; line integrals with respect to a scalar 11.2 11.3 11.4 11.5 Connectivity of regions Green's theorem in a plane Conservative ﬁelds and potentials Surface integrals 383 384 387 389 Evaluating surface integrals; vector areas of surfaces; physical examples 11.6 Volume integrals 396 Volumes of three-dimensional regions 11.7 11.8 Integral forms for grad, div and curl Divergence theorem and related theorems 398 401 Green's theorems; other related integral theorems; physical applications 11.9 Stokes' theorem and related theorems 406 Related integral theorems; physical applications 11.10 Exercises 11.11 Hints and answers 409 414 12 12.1 415 415 Fourier series The Dirichlet conditions ix CONTENTS 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 The Fourier coeﬃcients Symmetry considerations Discontinuous functions Non-periodic functions Integration and diﬀerentiation Complex Fourier series Parseval's theorem Exercises Hints and answers 417 419 420 422 424 424 426 427 431 13 13.1 Integral transforms Fourier transforms 433 433 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 13.2 Laplace transforms 453 Laplace transforms of derivatives and integrals; other properties of Laplace transforms 13.3 13.4 13.5 Concluding remarks Exercises Hints and answers 459 460 466 14 14.1 14.2 First-order ordinary diﬀerential equations General form of solution First-degree ﬁrst-order equations 468 469 470 Separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; isobaric equations; Bernoulli's equation; miscellaneous equations 14.3 Higher-degree ﬁrst-order equations 480 Equations soluble for p; for x; for y; Clairaut's equation 14.4 14.5 Exercises Hints and answers 484 488 15 15.1 Higher-order ordinary diﬀerential equations Linear equations with constant coeﬃcients 490 492 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 15.2 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 x 503 CONTENTS 15.3 General ordinary diﬀerential equations 518 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 15.4 15.5 Exercises Hints and answers 523 529 16 16.1 Series solutions of ordinary diﬀerential equations Second-order linear ordinary diﬀerential equations 531 531 Ordinary and singular points 16.2 16.3 Series solutions about an ordinary point Series solutions about a regular singular point 535 538 Distinct roots not diﬀering by an integer; repeated root of the indicial equation; distinct roots diﬀering by an integer 16.4 Obtaining a second solution 544 The Wronskian method; the derivative method; series form of the second solution 16.5 16.6 16.7 Polynomial solutions Exercises Hints and answers 548 550 553 17 17.1 Eigenfunction methods for diﬀerential equations Sets of functions 554 556 Some useful inequalities 17.2 17.3 Adjoint, self-adjoint and Hermitian operators Properties of Hermitian operators 559 561 Reality of the eigenvalues; orthogonality of the eigenfunctions; construction of real eigenfunctions 17.4 Sturm–Liouville equations 564 Valid boundary conditions; putting an equation into Sturm–Liouville form 17.5 17.6 17.7 17.8 Superposition of eigenfunctions: Green's functions A useful generalisation Exercises Hints and answers 569 572 573 576 18 18.1 Special functions Legendre functions 577 577 General solution for integer ; properties of Legendre polynomials 18.2 18.3 18.4 18.5 Associated Legendre functions Spherical harmonics Chebyshev functions Bessel functions 587 593 595 602 General solution for non-integer ν; general solution for integer ν; properties of Bessel functions xi CONTENTS 18.6 18.7 18.8 18.9 18.10 18.11 18.12 18.13 18.14 Spherical Bessel functions Laguerre functions Associated Laguerre functions Hermite functions Hypergeometric functions Conﬂuent hypergeometric functions The gamma function and related functions Exercises Hints and answers 614 616 621 624 628 633 635 640 646 19 19.1 Quantum operators Operator formalism 648 648 Commutators 19.2 Physical examples of operators 656 Uncertainty principle; angular momentum; creation and annihilation operators 19.3 19.4 Exercises Hints and answers 671 674 20 20.1 Partial diﬀerential equations: general and particular solutions Important partial diﬀerential equations 675 676 The wave equation; the diﬀusion equation; Laplace's equation; Poisson's equation; Schr¨odinger's equation 20.2 20.3 General form of solution General and particular solutions 680 681 First-order equations; inhomogeneous equations and problems; second-order equations 20.4 20.5 20.6 The wave equation The diﬀusion equation Characteristics and the existence of solutions 693 695 699 First-order equations; second-order equations 20.7 20.8 20.9 Uniqueness of solutions Exercises Hints and answers 705 707 711 21 Partial diﬀerential equations: separation of variables and other methods Separation of variables: the general method Superposition of separated solutions Separation of variables in polar coordinates 713 713 717 725 21.1 21.2 21.3 Laplace's equation in polar coordinates; spherical harmonics; other equations in polar coordinates; solution by expansion; separation of variables for inhomogeneous equations 21.4 Integral transform methods 747 xii CONTENTS 21.5 Inhomogeneous problems – Green's functions 751 Similarities to Green's functions for ordinary diﬀerential equations; general boundary-value problems; Dirichlet problems; Neumann problems 21.6 21.7 Exercises Hints and answers 767 773 22 22.1 22.2 Calculus of variations The Euler–Lagrange equation Special cases 775 776 777 F does not contain y explicitly; F does not contain x explicitly 22.3 Some extensions 781 Several dependent variables; several independent variables; higher-order derivatives; variable end-points 22.4 22.5 Constrained variation Physical variational principles 785 787 Fermat's principle in optics; Hamilton's principle in mechanics 22.6 22.7 22.8 22.9 22.10 General eigenvalue problems Estimation of eigenvalues and eigenfunctions Adjustment of parameters Exercises Hints and answers 790 792 795 797 801 23 23.1 23.2 23.3 23.4 Integral equations Obtaining an integral equation from a diﬀerential equation Types of integral equation Operator notation and the existence of solutions Closed-form solutions 803 803 804 805 806 Separable kernels; integral transform methods; diﬀerentiation 23.5 23.6 23.7 23.8 23.9 Neumann series Fredholm theory Schmidt–Hilbert theory Exercises Hints and answers 813 815 816 819 823 24 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 Complex variables Functions of a complex variable The Cauchy–Riemann relations Power series in a complex variable Some elementary functions Multivalued functions and branch cuts Singularities and zeros of complex functions Conformal transformations Complex integrals 824 825 827 830 832 835 837 839 845 xiii CONTENTS 24.9 24.10 24.11 24.12 24.13 24.14 24.15 Cauchy's theorem Cauchy's integral formula Taylor and Laurent series Residue theorem Deﬁnite integrals using contour integration Exercises Hints and answers 849 851 853 858 861 867 870 25 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 Applications of complex variables Complex potentials Applications of conformal transformations Location of zeros Summation of series Inverse Laplace transform Stokes' equation and Airy integrals WKB methods Approximations to integrals 871 871 876 879 882 884 888 895 905 Level lines and saddle points; steepest descents; stationary phase 25.9 Exercises 25.10 Hints and answers 920 925 26 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 26.10 26.11 26.12 26.13 26.14 26.15 26.16 26.17 26.18 26.19 26.20 927 928 929 930 932 935 938 939 941 944 946 949 950 954 955 957 960 963 965 968 971 Tensors Some notation Change of basis Cartesian tensors First- and zero-order Cartesian tensors Second- and higher-order Cartesian tensors The algebra of tensors The quotient law The tensors δij and ijk Isotropic tensors Improper rotations and pseudotensors Dual tensors Physical applications of tensors Integral theorems for tensors Non-Cartesian coordinates The metric tensor General coordinate transformations and tensors Relative tensors Derivatives of basis vector...
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