Riley K.F. Mathematical methods for physics and engineering / K.F.Riley, M.P.Hobson, S.J.Bence. - 3rd ed. - Cambridge; New York: Cambridge University Press, 2006. - xxvii, 1333 p.: ill. - Ind.: p.1305-1333. - ISBN 978-0-521-67971-8 Шифр: (И/В31-R57) 02  Место хранения:   02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents

Preface to the third edition ................................... xx Preface to the second edition ............................... xxiii Preface to the first edition .................................. xxv 1 Preliminary algebra .......................................... 1 1.1 Simple functions and equations .......................... 1 Polynomial equations; factorisation; properties of roots 1.2 Trigonometric identities ............................... 10 Single angle; compound angles; double- and half-angle identities 1.3 Coordinate geometry .................................... 15 1.4 Partial fractions ...................................... 18 Complications and special cases 1.5 Binomial expansion ..................................... 25 1.6 Properties of binomial coefficients .................... 27 1.7 Some particular methods of proof ....................... 30 Proof by induction; proof by contradiction; necessary and sufficient conditions 1.8 Exercises .............................................. 36 1.9 Hints and answers ...................................... 39 2 Preliminary calculus ........................................ 41 2.1 Differentiation ........................................ 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 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 Exercises .............................................. 76 2.4 Hints and answers ...................................... 81 3 Complex numbers and hyperbolic functions .................... 83 3.1 The need for complex numbers ........................... 83 3.2 Manipulation of complex numbers ........................ 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 Complex logarithms and complex powers .................. 99 3.6 Applications to differentiation and integration ....... 101 3.7 Hyperbolic functions .................................. 102 Definitions; hyperbolic-trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculus of hyperbolic functions 3.8 Exercises ............................................. 109 3.9 Hints and answers ..................................... 113 4 Series and limits .......................................... 115 4.1 Series ................................................ 115 4.2 Summation of series ................................... 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 Operations with series ................................ 131 4.5 Power series .......................................... 131 Convergence of power series; operations with power series 4.6 Taylor series ......................................... 136 Taylor's theorem; approximation errors; standard Maclaurin series 4.7 Evaluation of limits .................................. 141 4.8 Exercises ............................................. 144 4.9 Hints and answers ..................................... 149 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 of variables in double integrals; evaluation of the integral I = ∫∞-∞ e-x2dx; 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 7.7 Equations of lines, planes and spheres ................ 226 7.8 Using vectors to find distances ....................... 229 Point to line; point to plane; line to line; line to plane 7.9 Reciprocal vectors .................................... 233 7.10 Exercises ............................................. 234 7.11 Hints and answers ..................................... 240 8 Matrices and vector spaces ................................. 241 8.1 Vector spaces ......................................... 242 Basis vectors; inner product; some useful inequalities 8.2 Linear operators ...................................... 247 8.3 Matrices .............................................. 249 8.4 Basic matrix algebra .................................. 250 Matrix addition; multiplication by a scalar; matrix multiplication 8.5 Functions of matrices ................................. 255 8.6 The transpose of a matrix ............................. 255 8.7 The complex and Hermitian conjugates of a matrix ...... 256 8.8 The trace of a matrix ................................. 258 8.9 The determinant of a matrix ........................... 259 Properties of determinants 8.10 The inverse of a matrix ............................... 263 8.11 The rank of a matrix .................................. 267 8.12 Special types of square matrix ........................ 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 Change of basis and similarity transformations ........ 282 8.16 Diagonalisation of matrices ........................... 285 8.17 Quadratic and Hermitian forms ......................... 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 Exercises ............................................ 307 8.20 Hints and answers .................................... 314 9 Normal modes ............................................... 316 9.1 Typical oscillatory systems ........................... 317 9.2 Symmetry and normal modes ............................. 322 9.3 Rayleigh-Ritz method .................................. 327 9.4 Exercises ............................................. 329 9.5 Hints and answers ..................................... 332 10 Vector calculus ............................................ 334 10.1 Differentiation of vectors ............................ 334 Composite vector expressions; differential of a vector 10.2 Integration of vectors ................................ 339 10.3 Space curves .......................................... 340 10.4 Vector functions of several arguments ................. 344 10.5 Surfaces .............................................. 345 10.6 Scalar and vector fields .............................. 347 10.7 Vector operators ...................................... 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 Cylindrical and spherical polar coordinates ........... 357 10.10 General curvilinear coordinates ...................... 364 10.11 Exercises ............................................ 369 10.12 Hints and answers .................................... 375 11 Line, surface and volume integrals ......................... 377 11.1 Line integrals ........................................ 377 Evaluating line integrals; physical examples; line integrals with respect to a scalar 11.2 Connectivity of regions ............................... 383 11.3 Green's theorem in a plane ............................ 384 11.4 Conservative fields and potentials .................... 387 11.5 Surface integrals ..................................... 389 Evaluating surface integrals; vector areas of surfaces; physical examples 11.6 Volume integrals ...................................... 396 Volumes of three-dimensional regions 11.7 Integral forms for grad, div and curl ................. 398 11.8 Divergence theorem and related theorems ............... 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 ............................................ 409 11.11 Hints and answers .................................... 414 12 Fourier series ............................................. 415 12.1 The Dirichlet conditions ............................. 415 12.2 The Fourier coefficients ............................. 417 12.3 Symmetry considerations .............................. 419 12.4 Discontinuous functions .............................. 420 12.5 Non-periodic functions ............................... 422 12.6 Integration and differentiation ...................... 424 12.7 Complex Fourier series ............................... 424 12.8 Parseval's theorem ................................... 426 12.9 Exercises ............................................ 427 12.10 Hints and answers .................................... 431 13 Integral transforms ........................................ 433 13.1 Fourier transforms .................................... 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 Concluding remarks .................................... 459 13.4 Exercises ............................................. 460 13.5 Hints and answers ..................................... 466 14 First-order ordinary differential equations ................ 468 14.1 General form of solution .............................. 469 14.2 First-degree first-order equations .................... 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 Exercises ............................................. 484 14.5 Hints and answers ..................................... 488 15 Higher-order ordinary differential equations ............... 490 15.1 Linear equations with constant coefficients ........... 492 Finding the complementary function уc(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 ........... 503 The Legendre and Euler linear equations; exact equations; partially known complementary function; variation of parameters; Green's functions; canonical form for second-order equations 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 у alone; equations having у = Aex as a solution 15.4 Exercises ............................................. 523 15.5 Hints and answers ..................................... 529 16 Series solutions of ordinary differential equations ........ 531 16.1 Second-order linear ordinary differential equations ... 531 Ordinary and singular points 16.2 Series solutions about an ordinary point .............. 535 16.3 Series solutions about a regular singular point ....... 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 Polynomial solutions .................................. 548 16.6 Exercises ............................................. 550 16.7 Hints and answers ..................................... 553 17 Eigenfunction methods for differential equations ........... 554 17.1 Sets of functions ..................................... 556 Some useful inequalities 17.2 Adjoint, self-adjoint and Hermitian operators ......... 559 17.3 Properties of Hermitian operators ..................... 561 Reality of the eigenvalues; orthogonality of the eigenfunctions; construction of real eigenfunctions 14.4 Sturm-Liouville equations ............................. 564 Valid boundary conditions; putting an equation into Sturm-Liouville form 17.5 Superposition of eigenfunctions: Green's functions .... 569 17.6 A useful generalisation ............................... 572 17.7 Exercises ............................................. 573 17.8 Hints and answers ..................................... 576 18 Special functions .......................................... 577 18.1 Legendre functions .................................... 577 General solution for integer t; properties of Legendre polynomials 18.2 Associated Legendre functions ......................... 587 18.3 Spherical harmonics ................................... 593 18.4 Chebyshev functions ................................... 595 18.5 Bessel functions ...................................... 602 General solution for non-integer v; general solution for integer v; properties of Bessel functions 18.6 Spherical Bessel functions ............................ 614 18.7 Laguerre functions .................................... 616 18.8 Associated Laguerre functions ......................... 621 18.9 Hermite functions ..................................... 624 18.10 Hypergeometric functions ............................. 628 18.11 Confluent hypergeometric functions ................... 633 18.12 The gamma function and related functions ............. 635 18.13 Exercises ............................................ 640 18.14 Hints and answers .................................... 646 19 Quantum operators .......................................... 648 19.1 Operator formalism .................................... 648 Commutators 19.2 Physical examples of operators ........................ 656 Uncertainty principle; angular momentum; creation and annihilation operators 19.3 Exercises ............................................. 671 19.4 Hints and answers ..................................... 674 20 Partial differential equations: general and particular solutions .................................................. 675 20.1 Important partial differential equations .............. 676 The wave equation; the diffusion equation; Laplace's equation; Poisson's equation; Schrödinger 's equation 20.2 General form of solution .............................. 680 20.3 General and particular solutions ...................... 681 First-order equations; inhomogeneous equations and problems; second-order equations 20.4 The wave equation ..................................... 693 20.5 The diffusion equation ................................ 695 20.6 Characteristics and the existence of solutions ........ 699 First-order equations: second-order equations 20.7 Uniqueness of solutions ............................... 705 20.8 Exercises ............................................. 707 20.9 Hints and answers ..................................... 711 21 Partial differential equations: separation of variables and other methods .......................................... 713 21.1 Separation of variables: the general method ........... 713 21.2 Superposition of separated solutions .................. 717 21.3 Separation of variables in polar coordinates .......... 725 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 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 Exercises ............................................. 767 21.7 Hints and answers ..................................... 773 22 Calculus of variations ..................................... 775 22.1 The Euler-Lagrange equation ........................... 776 22.2 Special cases ......................................... 777 F does not contain у 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 Constrained variation ................................. 785 22.5 Physical variational principles ....................... 787 Fermat's principle in optics; Hamilton's principle in mechanics 22.6 General eigenvalue problems ........................... 790 22.7 Estimation of eigenvalues and eigenfunctions .......... 792 22.8 Adjustment of parameters .............................. 795 22.9 Exercises ............................................. 797 22.10 Hints and answers .................................... 801 23 Integral equations ......................................... 803 23.1 Obtaining an integral equation from a differential equation .............................................. 803 23.2 Types of integral equation ............................ 804 23.3 Operator notation and the existence of solutions ...... 805 23.4 Closed-form solutions ................................. 806 Separable kernels; integral transform methods; differentiation 23.5 Neumann series ........................................ 813 23.6 Fredholm theory ....................................... 815 23.7 Schmidt-Hilbert theory ................................ 816 23.8 Exercises ............................................. 819 23.9 Hints and answers ..................................... 823 24 Complex variables .......................................... 824 24.1 Functions of a complex variable ....................... 825 24.2 The Cauchy-Riemann relations .......................... 827 24.3 Power series in a complex variable .................... 830 24.4 Some elementary functions ............................. 832 24.5 Multivalued functions and branch cuts ................. 835 24.6 Singularities and zeros of complex functions .......... 837 24.7 Conformal transformations ............................. 839 24.8 Complex integrals ..................................... 845 24.9 Cauchy's theorem ...................................... 849 24.10 Cauchy's integral formula ............................ 851 24.11 Taylor and Laurent series ............................ 853 24.12 Residue theorem ...................................... 858 24.13 Definite integrals using contour integration ......... 861 24.14 Exercises ............................................ 867 24.15 Hints and answers .................................... 870 25 Applications of complex variables .......................... 871 25.1 Complex potentials .................................... 871 25.2 Applications of conformal transformations ............. 876 25.3 Location of zeros ..................................... 879 25.4 Summation of series ................................... 882 25.5 Inverse Laplace transform ............................. 884 25.6 Stokes' equation and Airy integrals ................... 888 25.7 WKB methods ........................................... 895 25.8 Approximations to integrals ........................... 905 Level lines and saddle points; steepest descents; stationary phase 25.9 Exercises ............................................. 920 25.10 Hints and answers .................................... 925 26 Tensors .................................................... 927 26.1 Some notation ......................................... 928 26.2 Change of basis ....................................... 929 26.3 Cartesian tensors ..................................... 930 26.4 First- and zero-order Cartesian tensors ............... 932 26.5 Second- and higher-order Cartesian tensors ............ 935 26.6 The algebra of tensors ................................ 938 26.7 The quotient law ...................................... 939 26.8 The tensors δij and єijk .............................. 941 26.9 Isotropic tensors ..................................... 944 26.10 Improper rotations and pseudotensors ................. 946 26.11 Dual tensors ......................................... 949 26.12 Physical applications of tensors ..................... 950 26.13 Integral theorems for tensors ........................ 954 26.14 Non-Cartesian coordinates ............................ 955 26.15 The metric tensor .................................... 957 26.16 General coordinate transformations and tensors ....... 960 26.17 Relative tensors ..................................... 963 26.18 Derivatives of basis vectors and Christoffel symbols .............................................. 965 26.19 Covariant differentiation ............................ 968 26.20 Vector operators in tensor form ...................... 971 26.21 Absolute derivatives along curves .................... 975 26.22 Geodesies ............................................ 976 26.23 Exercises ............................................ 977 26.24 Hints and answers .................................... 982 27 Numerical methods .......................................... 984 27.1 Algebraic and transcendental equations ................ 985 Rearrangement of the equation; linear interpolation; binary chopping; Newton-Raphson method 27.2 Convergence of iteration schemes ...................... 992 27.3 Simultaneous linear equations ......................... 994 Gaussian elimination; Gauss-Seidel iteration; tridiagonal matrices 27.4 Numerical integration ................................ 1000 Trapezium rule; Simpson's rule; Gaussian integration; Monte Carlo methods 27.5 Finite differences ................................... 1019 27.6 Differential equations ............................... 1020 Difference equations; Taylor series solutions; prediction and correction; Runge-Kutta methods; isoclines 27.7 Higher-order equations ............................... 1028 27.8 Partial differential equations ....................... 1030 27.9 Exercises ............................................ 1033 27.10 Hints and answers ................................... 1039 28 Group theory .............................................. 1041 28.1 Groups ............................................... 1041 Definition of a group; examples of groups 28.2 Finite groups ........................................ 1049 28.3 Non-Abelian groups ................................... 1052 28.4 Permutation groups ................................... 1056 28.5 Mappings between groups .............................. 1059 28.6 Subgroups ............................................ 1061 28.7 Subdividing a group .................................. 1063 Equivalence relations and classes; congruence and cosets; conjugates and classes 28.8 Exercises ............................................ 1070 28.9 Hints and answers .................................... 1074 29 Representation theory ..................................... 1076 29.1 Dipole moments of molecules .......................... 1077 29.2 Choosing an appropriate formalism .................... 1078 29.3 Equivalent representations ........................... 1084 29.4 Reducibility of a representation ..................... 1086 29.5 The orthogonality theorem for irreducible representations ...................................... 1090 29.6 Characters ........................................... 1092 Orthogonality property of characters 29.7 Counting irreps using characters ..................... 1095 Summation rules for irreps 29.8 Construction of a character table .................... 1100 29.9 Group nomenclature ................................... 1102 29.10 Product representations ............................. 1103 29.11 Physical applications of group theory ............... 1105 Bonding in molecules; matrix elements in quantum mechanics; degeneracy of normal modes; breaking of degeneracies 29.12 Exercises ........................................... 1113 29.13 Hints and answers ................................... 1117 30 Probability ............................................... 1119 30.1 Venn diagrams ........................................ 1119 30.2 Probability .......................................... 1124 Axioms and theorems; conditional probability; Bayes' theorem 30.3 Permutations and combinations ........................ 1133 30.4 Random variables and distributions ................... 1139 Discrete random variables; continuous random variables 30.5 Properties of distributions .......................... 1143 Mean: mode and median; variance and standard deviation: moments; central moments 30.6 Functions of random variables ........................ 1150 30.7 Generating functions ................................. 1157 Probability generating functions; moment generating functions; characteristic functions; cumulant generating functions 30.8 Important discrete distributions ..................... 1168 Binomial; geometric; negative binomial; hypergeometric; Poisson 30.9 Important continuous distributions ................... 1179 Gaussian; log-normal; exponential; gamma; chi-squared; Cauchy; Breit-Wigner; uniform 30.10 The central limit theorem ........................... 1195 30.11 Joint distributions ................................. 1196 Discrete bivariate; continuous bivariate; marginal and conditional distributions 30.12 Properties of joint distributions ................... 1199 Means; variances; covariance and correlation 30.13 Generating functions for joint distributions ........ 1205 30.14 Transformation of variables in joint distributions .. 1206 30.15 Important joint distributions ....................... 1207 Multinominal: multivariate Gaussian 30.16 Exercises ........................................... 1211 30.17 Hints and answers ................................... 1219 31 Statistics ................................................ 1221 31.1 Experiments, samples and populations ................. 1221 31.2 Sample statistics .................................... 1222 Averages; variance and standard deviation; moments; covariance and correlation 31.3 Estimators and sampling distributions ................ 1229 Consistency, bias and efficiency; Fisher's inequality; standard errors; confidence limits 31.4 Some basic estimators ................................ 1243 Mean; variance; standard deviation; moments; covariance and correlation 31.5 Maximum-likelihood method ............................ 1255 ML estimator; transformation invariance and bias; efficiency; errors and confidence limits; Bayesian interpretation; large-N behaviour: extended ML method 31.6 The method of least squares .......................... 1271 Linear least squares; non-linear least squares 31.7 Hypothesis testing ................................... 1277 Simple and composite hypotheses; statistical tests; Neyman-Pearson; generalised likelihood-ratio; Student's t; Fisher's F; goodness of fit 31.8 Exercises ............................................ 1298 31.9 Hints and answers .................................... 1303 Index ........................................................ 1305