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Mathematical methods for physics and engineering-[3rd ed]-[pdf]-[K. F. Riley, M. P. Hobson, S. J. Bence]

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    书籍信息:
    标题: Mathematical methods for physics and engineering
    语言: English
    格式: pdf
    大小: 9.3M
    页数: 1363
    年份: 2006
    作者: K. F. Riley, M. P. Hobson, S. J. Bence
    版次: 3rd ed
    出版社: Cambridge University Press

    简介

    The third edition of this highly acclaimed undergraduate textbook is suitable for teaching all the mathematics for an undergraduate course in any of the physical sciences. As well as lucid descriptions of all the topics and many worked examples, it contains over 800 exercises. 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, and give an introduction to quantum operators. Further tabulations, of relevance in statistics and numerical integration, have been added. In this edition, half of the exercises are provided with hints and answers and, in a separate manual available to both students and their teachers, complete worked solutions. The remaining exercises have no hints, answers or worked solutions and can be used for unaided homework; full solutions are available to instructors on a password-protected web site, www.cambridge.org/9780521679718.


    目录
    Cover Page......Page 1
    Mathematical Methods for Physics and Engineering......Page 3
    Title Page......Page 5
    ISBN 0521861535......Page 6
    2 Preliminary calculus......Page 7
    4 Series and limits......Page 8
    7 Vector algebra......Page 9
    9 Normal modes......Page 10
    12 Fourier series......Page 11
    15 Higher-order ordinary differential equations......Page 12
    18 Special functions......Page 13
    21 Partial differential equations: separation of variables and other methods......Page 14
    24 Complex variables......Page 15
    26 Tensors......Page 16
    29 Representation theory......Page 17
    30 Probability......Page 18
    31 Statistics......Page 19
    Preface to the third edition......Page 22
    Preface to the second edition......Page 25
    Preface to the first edition......Page 27
    1.1.1 Polynomials and polynomial equations......Page 31
    1.1.2 Factorising polynomials......Page 37
    1.1.3 Properties of roots......Page 39
    1.2.1 Single-angle identities......Page 40
    1.2.2 Compound-angle identities......Page 41
    1.2.3 Double- and half-angle identities......Page 43
    1.3 Coordinate geometry......Page 45
    1.4 Partial fractions......Page 48
    1.4.1 Complications and special cases......Page 51
    1.5 Binomial expansion......Page 55
    1.5.2 Proof of the binomial expansion......Page 56
    1.6.1 Identities involving binomial coefficients......Page 57
    1.6.2 Negative and non-integral values of......Page 59
    1.7 Some particular methods of proof......Page 60
    1.7.1 Proof by induction......Page 61
    1.7.2 Proof by contradiction......Page 62
    1.7.3 Necessary and sufficient conditions......Page 64
    1.8 Exercises......Page 66
    1.9 Hints and answers......Page 69
    2.1.1 Differentiation from first principles......Page 71
    2.1.2 Differentiation of products......Page 74
    2.1.3 The chain rule......Page 76
    2.1.5 Implicit differentiation......Page 77
    2.1.7 Leibnitz’ theorem......Page 78
    2.1.8 Special points of a function......Page 80
    2.1.9 Curvature of a function......Page 82
    2.1.10 Theorems of differentiation......Page 85
    2.2.1 Integration from first principles......Page 89
    2.2.2 Integration as the inverse of differentiation......Page 91
    2.2.3 Integration by inspection......Page 92
    2.2.4 Integration of sinusoidal functions......Page 93
    2.2.6 Integration using partial fractions......Page 94
    2.2.7 Integration by substitution......Page 95
    2.2.8 Integration by parts......Page 97
    2.2.9 Reduction formulae......Page 99
    2.2.11 Integration in plane polar coordinates......Page 100
    2.2.13 Applications of integration......Page 102
    2.3 Exercises......Page 106
    2.4 Hints and answers......Page 111
    3.1 The need for complex numbers......Page 113
    3.2.1 Addition and subtraction......Page 115
    3.2.2 Modulus and argument......Page 117
    3.2.3 Multiplication......Page 118
    3.2.4 Complex conjugate......Page 119
    3.2.5 Division......Page 121
    3.3 Polar representation of complex numbers......Page 122
    3.3.1 Multiplication and division in polar form......Page 124
    3.4.1 Trigonometric identities......Page 125
    3.4.2 Finding the nth roots of unity......Page 127
    3.4.3 Solving polynomial equations......Page 128
    3.5 Complex logarithms and complex powers......Page 129
    3.6 Applications to differentiation and integration......Page 131
    3.7.2 Hyperbolic–trigonometric analogies......Page 132
    3.7.3 Identities of hyperbolic functions......Page 134
    3.7.5 Inverses of hyperbolic functions......Page 135
    3.7.6 Calculus of hyperbolic functions......Page 136
    3.8 Exercises......Page 139
    3.9 Hints and answers......Page 143
    4.1 Series......Page 145
    4.2 Summation of series......Page 146
    4.2.2 Geometric series......Page 147
    4.2.3 Arithmetico-geometric series......Page 148
    4.2.4 The difference method......Page 149
    4.2.5 Series involving natural numbers......Page 151
    4.2.6 Transformation of series......Page 152
    4.3.1 Absolute and conditional convergence......Page 154
    4.3.2 Convergence of a series containing only real positive terms......Page 155
    4.3.3 Alternating series test......Page 160
    4.5 Power series......Page 161
    4.5.1 Convergence of power series......Page 162
    4.5.2 Operations with power series......Page 164
    4.6.1 Taylor’s theorem......Page 166
    4.6.2 Approximation errors in Taylor series......Page 169
    4.6.3 Standard Maclaurin series......Page 170
    4.7 Evaluation of limits......Page 171
    4.8 Exercises......Page 174
    4.9 Hints and answers......Page 179
    5.1 Definition of the partial derivative......Page 181
    5.2 The total differential and total derivative......Page 183
    5.3 Exact and inexact differentials......Page 185
    5.5 The chain rule......Page 187
    5.6 Change of variables......Page 188
    5.7 Taylor’s theorem for many-variable functions......Page 190
    5.8 Stationary values of many-variable functions......Page 192
    5.9 Stationary values under constraints......Page 197
    5.10 Envelopes......Page 203
    5.10.1 Envelope equations......Page 204
    5.11 Thermodynamic relations......Page 206
    5.12 Differentiation of integrals......Page 208
    5.13 Exercises......Page 209
    5.14 Hints and answers......Page 215
    6.1 Double integrals......Page 217
    6.2 Triple integrals......Page 220
    6.3.1 Areas and volumes......Page 221
    6.3.2 Masses, centres of mass and centroids......Page 223
    6.3.3 Pappus’ theorems......Page 225
    6.3.4 Moments of inertia......Page 228
    6.4 Change of variables in multiple integrals......Page 229
    6.4.1 Change of variables in double integrals......Page 230
    6.4.2 Evaluation of the integral......Page 232
    6.4.3 Change of variables in triple integrals......Page 234
    6.4.4 General properties of Jacobians......Page 236
    6.5 Exercises......Page 237
    6.6 Hints and answers......Page 241
    7.1 Scalars and vectors......Page 242
    7.2 Addition and subtraction of vectors......Page 243
    7.3 Multiplication by a scalar......Page 244
    7.4 Basis vectors and components......Page 247
    7.5 Magnitude of a vector......Page 248
    7.6.1 Scalar product......Page 249
    7.6.2 Vector product......Page 252
    7.6.3 Scalar triple product......Page 254
    7.7.1 Equation of a line......Page 256
    7.7.2 Equation of a plane......Page 257
    7.7.3 Equation of a sphere......Page 258
    7.8.1 Distance from a point to a line......Page 259
    7.8.2 Distance from a point to a plane......Page 260
    7.8.3 Distance from a line to a line......Page 261
    7.8.4 Distance from a line to a plane......Page 262
    7.9 Reciprocal vectors......Page 263
    7.10 Exercises......Page 264
    7.11 Hints and answers......Page 270
    8 Matrices and vector spaces......Page 271
    8.1 Vector spaces......Page 272
    8.1.1 Basis vectors......Page 273
    8.1.2 The inner product......Page 274
    8.1.3 Some useful inequalities......Page 276
    8.2 Linear operators......Page 277
    8.3 Matrices......Page 279
    8.4 Basic matrix algebra......Page 280
    8.4.1 Matrix addition and multiplication by a scalar......Page 281
    8.4.2 Multiplication of matrices......Page 282
    8.4.3 The null and identity matrices......Page 284
    8.6 The transpose of a matrix......Page 285
    8.7 The complex and Hermitian conjugates of a matrix......Page 286
    8.8 The trace of a matrix......Page 288
    8.9 The determinant of a matrix......Page 289
    8.9.1 Properties of determinants......Page 291
    8.10 The inverse of a matrix......Page 293
    8.11 The rank of a matrix......Page 297
    8.12.1 Diagonal matrices......Page 298
    8.12.2 Lower and upper triangular matrices......Page 299
    8.12.4 Orthogonal matrices......Page 300
    8.12.6 Unitary matrices......Page 301
    8.13 Eigenvectors and eigenvalues......Page 302
    8.13.1 Eigenvectors and eigenvalues of a normal matrix......Page 303
    8.13.2 Eigenvectors and eigenvalues of Hermitian and anti-Hermitian matrices......Page 306
    8.13.5 Simultaneous eigenvectors......Page 308
    8.14 Determination of eigenvalues and eigenvectors......Page 310
    8.14.1 Degenerate eigenvalues......Page 311
    8.15 Change of basis and similarity transformations......Page 312
    8.16 Diagonalisation of matrices......Page 315
    8.17 Quadratic and Hermitian forms......Page 318
    8.17.1 The stationary properties of the eigenvectors......Page 320
    8.18 Simultaneous linear equations......Page 322
    8.18.1 The range and null space of a matrix......Page 323
    8.18.2 N simultaneous linear equations in N unknowns......Page 325
    8.18.3 Singular value decomposition......Page 331
    8.19 Exercises......Page 337
    8.20 Hints and answers......Page 344
    9 Normal modes......Page 346
    9.1 Typical oscillatory systems......Page 347
    9.2 Symmetry and normal modes......Page 352
    9.3 Rayleigh–Ritz method......Page 357
    9.4 Exercises......Page 359
    9.5 Hints and answers......Page 362
    10.1 Differentiation of vectors......Page 364
    10.1.1 Differentiation of composite vector expressions......Page 367
    10.1.2 Differential of a vector......Page 368
    10.2 Integration of vectors......Page 369
    10.3 Space curves......Page 370
    10.4 Vector functions of several arguments......Page 374
    10.5 Surfaces......Page 375
    10.7 Vector operators......Page 377
    10.7.1 Gradient of a scalar field......Page 378
    10.7.2 Divergence of a vector field......Page 382
    10.7.3 Curl of a vector field......Page 383
    10.8.1 Vector operators acting on sums and products......Page 384
    10.8.2 Combinations of grad, div and curl......Page 385
    10.9.1 Cylindrical polar coordinates......Page 387
    10.9.2 Spherical polar coordinates......Page 391
    10.10 General curvilinear coordinates......Page 394
    10.11 Exercises......Page 399
    10.12 Hints and answers......Page 405
    11.1 Line integrals......Page 407
    11.1.1 Evaluating line integrals......Page 408
    11.1.2 Physical examples of line integrals......Page 411
    11.1.3 Line integrals with respect to a scalar......Page 412
    11.2 Connectivity of regions......Page 413
    11.3 Green’s theorem in a plane......Page 414
    11.4 Conservative fields and potentials......Page 417
    11.5 Surface integrals......Page 419
    11.5.1 Evaluating surface integrals......Page 421
    11.5.2 Vector areas of surfaces......Page 423
    11.5.3 Physical examples of surface integrals......Page 425
    11.6 Volume integrals......Page 426
    11.6.1 Volumes of three-dimensional regions......Page 427
    11.7 Integral forms for grad, div and curl......Page 428
    11.8 Divergence theorem and related theorems......Page 431
    11.8.1 Green’s theorems......Page 432
    11.8.2 Other related integral theorems......Page 433
    11.8.3 Physical applications of the divergence theorem......Page 434
    11.9 Stokes’ theorem and related theorems......Page 436
    11.9.1 Related integral theorems......Page 437
    11.9.2 Physical applications of Stokes’ theorem......Page 438
    11.10 Exercises......Page 439
    11.11 Hints and answers......Page 444
    12.1 The Dirichlet conditions......Page 445
    12.2 The Fourier coefficients......Page 447
    12.3 Symmetry considerations......Page 449
    12.4 Discontinuous functions......Page 450
    12.5 Non-periodic functions......Page 452
    12.7 Complex Fourier series......Page 454
    12.8 Parseval’s theorem......Page 456
    12.9 Exercises......Page 457
    12.10 Hints and answers......Page 461
    13.1 Fourier transforms......Page 463
    13.1.1 The uncertainty principle......Page 465
    13.1.2 Fraunhofer diffraction......Page 467
    13.1.3 The Dirac δ-function......Page 469
    13.1.4 Relation of the δ-function to Fourier transforms......Page 472
    13.1.5 Properties of Fourier transforms......Page 473
    13.1.6 Odd and even functions......Page 475
    13.1.7 Convolution and deconvolution......Page 476
    13.1.8 Correlation functions and energy spectra......Page 479
    13.1.9 Parseval’s theorem......Page 480
    13.1.10 Fourier transforms in higher dimensions......Page 481
    13.2 Laplace transforms......Page 483
    13.2.1 Laplace transforms of derivatives and integrals......Page 485
    13.2.2 Other properties of Laplace transforms......Page 486
    13.3 Concluding remarks......Page 489
    13.4 Exercises......Page 490
    13.5 Hints and answers......Page 496
    14 First-order ordinary differential equations......Page 498
    14.1 General form of solution......Page 499
    14.2 First-degree first-order equations......Page 500
    14.2.1 Separable-variable equations......Page 501
    14.2.2 Exact equations......Page 502
    14.2.3 Inexact equations: integrating factors......Page 503
    14.2.4 Linear equations......Page 504
    14.2.5 Homogeneous equations......Page 505
    14.2.6 Isobaric equations......Page 506
    14.2.7 Bernoulli’s equation......Page 507
    14.2.8 Miscellaneous equations......Page 508
    14.3.1 Equations soluble for......Page 510
    14.3.2 Equations soluble for......Page 511
    14.3.3 Equations soluble for......Page 512
    14.3.4 Clairaut’s equation......Page 513
    14.4 Exercises......Page 514
    14.5 Hints and answers......Page 518
    15 Higher-order ordinary differential equations......Page 520
    15.1.1 Finding the complementary function......Page 522
    15.1.2 Finding the particular integral......Page 524
    15.1.3 Constructing the general solution......Page 525
    15.1.4 Linear recurrence relations......Page 526
    15.1.5 Laplace transform method......Page 531
    15.2.1 The Legendre and Euler linear equations......Page 533
    15.2.2 Exact equations......Page 535
    15.2.3 Partially known complementary function......Page 536
    15.2.4 Variation of parameters......Page 538
    15.2.5 Green’s functions......Page 541
    15.2.6 Canonical form for second-order equations......Page 546
    15.3.2 Independent variable absent......Page 548
    15.3.3 Non-linear exact equations......Page 549
    15.3.4 Isobaric or homogeneous equations......Page 551
    15.3.5 Equations homogeneous in......Page 552
    15.4 Exercises......Page 553
    15.5 Hints and answers......Page 559
    16.1 Second-order linear ordinary differential equations......Page 561
    16.1.1 Ordinary and singular points of an ODE......Page 563
    16.2 Series solutions about an ordinary point......Page 565
    16.3 Series solutions about a regular singular point......Page 568
    16.3.1 Distinct roots not differing by an integer......Page 570
    16.3.3 Distinct roots differing by an integer......Page 572
    16.4.1 The Wronskian method......Page 574
    16.4.2 The derivative method......Page 575
    16.4.3 Series form of the second solution......Page 577
    16.5 Polynomial solutions......Page 578
    16.6 Exercises......Page 580
    16.7 Hints and answers......Page 583
    17 Eigenfunction methods for differential equations......Page 584
    17.1 Sets of functions......Page 586
    17.2 Adjoint, self-adjoint and Hermitian operators......Page 589
    17.3.2 Orthogonality and normalisation of the eigenfunctions......Page 591
    17.3.4 Construction of real eigenfunctions......Page 593
    17.4.1 Hermitian nature of the Sturm–Liouville operator......Page 594
    17.4.2 Transforming an equation into Sturm–Liouville form......Page 595
    17.5 Superposition of eigenfunctions: Green’s functions......Page 599
    17.6 A useful generalisation......Page 602
    17.7 Exercises......Page 603
    17.8 Hints and answers......Page 606
    18.1 Legendre functions......Page 607
    18.1.1 Legendre functions for integer......Page 608
    18.1.2 Properties of Legendre polynomials......Page 610
    18.2 Associated Legendre functions......Page 617
    18.2.1 Associated Legendre functions for integer......Page 618
    18.2.2 Properties of associated Legendre functions Pm......Page 619
    18.3 Spherical harmonics......Page 623
    18.4 Chebyshev functions......Page 625
    18.4.1 Properties of Chebyshev polynomials......Page 629
    18.5 Bessel functions......Page 632
    18.5.1 Bessel functions for non-integer......Page 633
    18.5.2 Bessel functions for integer......Page 635
    18.5.3 Properties of Bessel functions......Page 638
    18.6 Spherical Bessel functions......Page 644
    18.7 Laguerre functions......Page 646
    18.7.1 Properties of Laguerre polynomials......Page 648
    18.8 Associated Laguerre functions......Page 651
    18.8.1 Properties of associated Laguerre polynomials......Page 652
    18.9 Hermite functions......Page 654
    18.9.1 Properties of Hermite polynomials......Page 656
    18.10 Hypergeometric functions......Page 658
    18.10.1 Properties of hypergeometric functions......Page 660
    18.11 Confluent hypergeometric functions......Page 663
    18.11.1 Properties of confluent hypergeometric functions......Page 664
    18.12 The gamma function and related functions......Page 665
    18.12.1 The gamma function......Page 666
    18.12.2 The beta function......Page 668
    18.12.3 The incomplete gamma function......Page 669
    18.13 Exercises......Page 670
    18.14 Hints and answers......Page 676
    19.1 Operator formalism......Page 678
    19.1.1 Commutation and commutators......Page 682
    19.2 Physical examples of operators......Page 686
    19.2.1 Angular momentum operators......Page 688
    19.2.2 Uncertainty principles......Page 694
    19.2.3 Annihilation and creation operators......Page 697
    19.3 Exercises......Page 701
    19.4 Hints and answers......Page 704
    20 Partial differential equations: general and particular solutions......Page 705
    20.1.1 The wave equation......Page 706
    20.1.2 The diffusion equation......Page 708
    20.1.5 Schr-odinger’s equation......Page 709
    20.2 General form of solution......Page 710
    20.3.1 First-order equations......Page 711
    20.3.2 Inhomogeneous equations and problems......Page 715
    20.3.3 Second-order equations......Page 717
    20.4 The wave equation......Page 723
    20.5 The diffusion equation......Page 725
    20.6.1 First-order equations......Page 729
    20.6.2 Second-order equations......Page 731
    20.7 Uniqueness of solutions......Page 735
    20.8 Exercises......Page 737
    20.9 Hints and answers......Page 741
    21.1 Separation of variables: the general method......Page 743
    21.2 Superposition of separated solutions......Page 747
    21.3.1 Laplace’s equation in polar coordinates......Page 755
    21.3.2 Other equations in polar coordinates......Page 767
    21.3.3 Solution by expansion......Page 771
    21.3.4 Separation of variables for inhomogeneous equations......Page 774
    21.4 Integral transform methods......Page 777
    21.5 Inhomogeneous problems – Green’s functions......Page 781
    21.5.1 Similarities to Green’s functions for ODEs......Page 782
    21.5.2 General boundary-value problems......Page 783
    21.5.3 Dirichlet problems......Page 786
    21.5.4 Neumann problems......Page 795
    21.6 Exercises......Page 797
    21.7 Hints and answers......Page 803
    22 Calculus of variations......Page 805
    22.1 The Euler–Lagrange equation......Page 806
    22.2.1 F does not contain y explicitly......Page 807
    22.2.2 F does not contain x explicitly......Page 809
    22.3 Some extensions......Page 811
    22.3.4 Variable end-points......Page 812
    22.4 Constrained variation......Page 815
    22.5.1 Fermat’s principle in optics......Page 817
    22.5.2 Hamilton’s principle in mechanics......Page 818
    22.6 General eigenvalue problems......Page 820
    22.7 Estimation of eigenvalues and eigenfunctions......Page 822
    22.8 Adjustment of parameters......Page 825
    22.9 Exercises......Page 827
    22.10 Hints and answers......Page 831
    23.1 Obtaining an integral equation from a differential equation......Page 833
    23.2 Types of integral equation......Page 834
    23.3 Operator notation and the existence of solutions......Page 835
    23.4 Closed-form solutions......Page 836
    23.4.1 Separable kernels......Page 837
    23.4.2 Integral transform methods......Page 839
    23.4.3 Differentiation......Page 842
    23.5 Neumann series......Page 843
    23.6 Fredholm theory......Page 845
    23.7 Schmidt–Hilbert theory......Page 846
    23.8 Exercises......Page 849
    23.9 Hints and answers......Page 853
    24 Complex variables......Page 854
    24.1 Functions of a complex variable......Page 855
    24.2 The Cauchy–Riemann relations......Page 857
    24.3 Power series in a complex variable......Page 860
    24.4 Some elementary functions......Page 862
    24.5 Multivalued functions and branch cuts......Page 865
    24.6 Singularities and zeros of complex functions......Page 867
    24.7 Conformal transformations......Page 869
    24.8 Complex integrals......Page 875
    24.9 Cauchy’s theorem......Page 879
    24.10 Cauchy’s integral formula......Page 881
    24.11 Taylor and Laurent series......Page 883
    24.12 Residue theorem......Page 888
    24.13.1 Integrals of sinusoidal functions......Page 891
    24.13.2 Some infinite integrals......Page 892
    24.13.3 Integrals of multivalued functions......Page 895
    24.14 Exercises......Page 897
    24.15 Hints and answers......Page 900
    25.1 Complex potentials......Page 901
    25.2 Applications of conformal transformations......Page 906
    25.3 Location of zeros......Page 909
    25.4 Summation of series......Page 912
    25.5 Inverse Laplace transform......Page 914
    25.6.1 The solutions of Stokes’ equation......Page 918
    25.6.3 Contour integral solutions......Page 920
    25.7.1 Phase memory......Page 925
    25.7.2 Constructing the WKB solutions......Page 927
    25.7.3 Accuracy of the WKB solutions......Page 932
    25.7.4 The Stokes phenomenon......Page 933
    25.8.1 Level lines and saddle points......Page 935
    25.8.2 Steepest descents method......Page 938
    25.8.3 Stationary phase method......Page 942
    25.9 Exercises......Page 950
    25.10 Hints and answers......Page 955
    26 Tensors......Page 957
    26.1 Some notation......Page 958
    26.2 Change of basis......Page 959
    26.3 Cartesian tensors......Page 960
    26.4 First- and zero-order Cartesian tensors......Page 962
    26.5 Second- and higher-order Cartesian tensors......Page 965
    26.6 The algebra of tensors......Page 968
    26.7 The quotient law......Page 969
    26.8 The tensors δij and ......Page 971
    26.9 Isotropic tensors......Page 974
    26.10 Improper rotations and pseudotensors......Page 976
    26.11 Dual tensors......Page 979
    26.12 Physical applications of tensors......Page 980
    26.13 Integral theorems for tensors......Page 984
    26.14 Non-Cartesian coordinates......Page 985
    26.15 The metric tensor......Page 987
    26.16 General coordinate transformations and tensors......Page 990
    26.17 Relative tensors......Page 993
    26.18 Derivatives of basis vectors and Christoffel symbols......Page 995
    26.19 Covariant differentiation......Page 998
    26.20 Vector operators in tensor form......Page 1001
    26.21 Absolute derivatives along curves......Page 1005
    26.22 Geodesics......Page 1006
    26.23 Exercises......Page 1007
    26.24 Hints and answers......Page 1012
    27 Numerical methods......Page 1014
    27.1 Algebraic and transcendental equations......Page 1015
    27.1.1 Rearrangement of the equation......Page 1017
    27.1.2 Linear interpolation......Page 1018
    27.1.4 Newton–Raphson method......Page 1020
    27.2 Convergence of iteration schemes......Page 1022
    27.3 Simultaneous linear equations......Page 1024
    27.3.1 Gaussian elimination......Page 1025
    27.3.2 Gauss–Seidel iteration......Page 1026
    27.3.3 Tridiagonal matrices......Page 1028
    27.4 Numerical integration......Page 1030
    27.4.1 Trapezium rule......Page 1032
    27.4.2 Simpson’s rule......Page 1034
    27.4.3 Gaussian integration......Page 1035
    27.4.4 Monte Carlo methods......Page 1039
    27.5 Finite differences......Page 1049
    27.6 Differential equations......Page 1050
    27.6.1 Difference equations......Page 1051
    27.6.2 Taylor series solutions......Page 1053
    27.6.3 Prediction and correction......Page 1054
    27.6.4 Runge–Kutta methods......Page 1056
    27.7 Higher-order equations......Page 1058
    27.8 Partial differential equations......Page 1060
    27.9 Exercises......Page 1063
    27.10 Hints and answers......Page 1069
    28.1 Groups......Page 1071
    28.1.1 Definition of a group......Page 1073
    28.1.2 Further examples of groups......Page 1078
    28.2 Finite groups......Page 1079
    28.3 Non-Abelian groups......Page 1082
    28.4 Permutation groups......Page 1086
    28.5 Mappings between groups......Page 1089
    28.6 Subgroups......Page 1091
    28.7 Subdividing a group......Page 1093
    28.7.1 Equivalence relations and classes......Page 1094
    28.7.2 Congruence and cosets......Page 1095
    28.7.3 Conjugates and classes......Page 1098
    28.8 Exercises......Page 1100
    28.9 Hints and answers......Page 1104
    29 Representation theory......Page 1106
    29.1 Dipole moments of molecules......Page 1107
    29.2 Choosing an appropriate formalism......Page 1108
    29.3 Equivalent representations......Page 1114
    29.4 Reducibility of a representation......Page 1116
    29.5 The orthogonality theorem for irreducible representations......Page 1120
    29.6 Characters......Page 1122
    29.6.1 Orthogonality property of characters......Page 1124
    29.7 Counting irreps using characters......Page 1125
    29.7.1 Summation rules for irreps......Page 1127
    29.8 Construction of a character table......Page 1130
    29.9 Group nomenclature......Page 1132
    29.10 Product representations......Page 1133
    29.11 Physical applications of group theory......Page 1135
    29.11.1 Bonding in molecules......Page 1136
    29.11.2 Matrix elements in quantum mechanics......Page 1138
    29.11.3 Degeneracy of normal modes......Page 1140
    29.11.4 Breaking of degeneracies......Page 1141
    29.12 Exercises......Page 1143
    29.13 Hints and answers......Page 1147
    30.1 Venn diagrams......Page 1149
    30.2 Probability......Page 1154
    30.2.1 Axioms and theorems......Page 1155
    30.2.2 Conditional probability......Page 1158
    30.2.3 Bayes’ theorem......Page 1162
    30.3.1 Permutations......Page 1163
    30.3.2 Combinations......Page 1165
    30.4.1 Discrete random variables......Page 1169
    30.4.2 Continuous random variables......Page 1170
    30.4.3 Sets of random variables......Page 1172
    30.5 Properties of distributions......Page 1173
    30.5.1 Mean......Page 1174
    30.5.2 Mode and median......Page 1175
    30.5.3 Variance and standard deviation......Page 1176
    30.5.4 Moments......Page 1177
    30.5.5 Central moments......Page 1178
    30.6 Functions of random variables......Page 1180
    30.6.2 Continuous random variables......Page 1181
    30.6.3 Functions of several random variables......Page 1183
    30.6.4 Expectation values and variances......Page 1185
    30.7.1 Probability generating functions......Page 1187
    30.7.2 Moment generating functions......Page 1192
    30.7.3 Characteristic function......Page 1195
    30.7.4 Cumulant generating function......Page 1196
    30.8.1 The binomial distribution......Page 1198
    30.8.2 The geometric and negative binomial distributions......Page 1202
    30.8.3 The hypergeometric distribution......Page 1203
    30.8.4 The Poisson distribution......Page 1204
    30.9.1 The Gaussian distribution......Page 1209
    30.9.3 The exponential and gamma distributions......Page 1220
    30.9.5 The Cauchy and Breit–Wigner distributions......Page 1223
    30.9.6 The uniform distribution......Page 1224
    30.10 The central limit theorem......Page 1225
    30.11 Joint distributions......Page 1226
    30.11.1 Discrete bivariate distributions......Page 1227
    30.11.2 Continuous bivariate distributions......Page 1228
    30.12.1 Means......Page 1229
    30.12.3 Covariance and correlation......Page 1230
    30.13 Generating functions for joint distributions......Page 1235
    30.14 Transformation of variables in joint distributions......Page 1236
    30.15 Important joint distributions......Page 1237
    30.15.1 The multinomial distribution......Page 1238
    30.15.2 The multivariate Gaussian distribution......Page 1239
    30.16 Exercises......Page 1241
    30.17 Hints and answers......Page 1249
    31.1 Experiments, samples and populations......Page 1251
    31.2 Sample statistics......Page 1252
    31.2.1 Averages......Page 1253
    31.2.2 Variance and standard deviation......Page 1254
    31.2.3 Moments and central moments......Page 1256
    31.2.4 Covariance and correlation......Page 1257
    31.3 Estimators and sampling distributions......Page 1259
    31.3.1 Consistency, bias and efficiency of estimators......Page 1260
    31.3.2 Fisher’s inequality......Page 1263
    31.3.3 Standard errors on estimators......Page 1264
    31.3.4 Confidence limits on estimators......Page 1265
    31.3.5 Confidence limits for a Gaussian sampling distribution......Page 1268
    31.3.6 Estimation of several quantities simultaneously......Page 1269
    31.4.1 Population mean......Page 1273
    31.4.2 Population variance σ2......Page 1275
    31.4.3 Population standard deviation......Page 1278
    31.4.4 Population moments......Page 1279
    31.4.5 Population central moments......Page 1280
    31.4.6 Population covariance Cov[x, y] and correlation Corr[x, y]......Page 1282
    31.4.7 A worked example......Page 1284
    31.5 Maximum-likelihood method......Page 1285
    31.5.1 The maximum-likelihood estimator......Page 1287
    31.5.2 Transformation invariance and bias of ML estimators......Page 1291
    31.5.3 Efficiency of ML estimators......Page 1292
    31.5.4 Standard errors and confidence limits on ML estimators......Page 1293
    31.5.5 The Bayesian interpretation of the likelihood function......Page 1294
    31.5.6 Behaviour of ML estimators for large......Page 1299
    31.5.7 Extended maximum-likelihood method......Page 1300
    31.6 The method of least squares......Page 1301
    31.6.1 Linear least squares......Page 1302
    31.6.2 Non-linear least squares......Page 1306
    31.7 Hypothesis testing......Page 1307
    31.7.2 Statistical tests......Page 1308
    31.7.3 The Neyman–Pearson test......Page 1310
    31.7.4 The generalised likelihood-ratio test......Page 1311
    31.7.5 Student’s t-test......Page 1314
    31.7.6 Fisher’s F-test......Page 1320
    31.7.7 Goodness of fit in least-squares problems......Page 1326
    31.8 Exercises......Page 1328
    31.9 Hints and answers......Page 1333
    A......Page 1335
    B......Page 1336
    C......Page 1337
    D......Page 1341
    E......Page 1343
    F......Page 1344
    G......Page 1345
    H......Page 1347
    I......Page 1348
    J,K......Page 1349
    L......Page 1350
    M......Page 1351
    N......Page 1353
    O......Page 1354
    P......Page 1355
    R......Page 1357
    S......Page 1358
    V......Page 1361
    Z......Page 1363

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