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ОбложкаRiley 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

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