Essential Mathematical Methods for the Physical Sciences

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 15
Review of background topics......Page 18
1 Matrices and vector spaces......Page 19
1.1 Vector spaces......Page 20
1.1.2 The inner product......Page 21
1.2 Linear operators......Page 23
1.3 Matrices......Page 25
1.4 Basic matrix algebra......Page 26
1.4.1 Matrix addition and multiplication by a scalar......Page 27
1.4.2 Multiplication of matrices......Page 28
1.4.3 The null and identity matrices......Page 30
1.6 The transpose of a matrix......Page 31
1.7 The complex and Hermitian conjugates of a matrix......Page 32
1.8 The trace of a matrix......Page 34
1.9 The determinant of a matrix......Page 35
1.9.1 Properties of determinants......Page 37
1.9.2 Evaluation of determinants......Page 38
1.10 The inverse of a matrix......Page 39
1.11 The rank of a matrix......Page 43
No solution......Page 45
1.12.2 N simultaneous linear equations in N unknowns......Page 46
Gaussian elimination......Page 47
Direct inversion......Page 48
LU decomposition......Page 49
Cramer’s rule......Page 52
1.12.3 A geometrical interpretation......Page 53
1.13 Special types of square matrix......Page 54
1.13.2 Lower and upper triangular matrices......Page 55
1.13.4 Orthogonal matrices......Page 56
1.13.6 Unitary matrices......Page 57
1.14 Eigenvectors and eigenvalues......Page 58
1.14.1 Eigenvectors and eigenvalues of Hermitian and unitary matrices......Page 60
1.14.3 Simultaneous eigenvectors......Page 62
1.15 Determination of eigenvalues and eigenvectors......Page 63
1.15.1 Degenerate eigenvalues......Page 65
1.16 Change of basis and similarity transformations......Page 67
1.17 Diagonalization of matrices......Page 69
1.18 Quadratic and Hermitian forms......Page 71
1.18.1 The stationary properties of the eigenvectors......Page 73
1.18.2 Quadratic surfaces......Page 75
1.19 Normal modes......Page 76
1.19.1 Typical oscillatory systems......Page 77
1.19.2 Rayleigh--Ritz method......Page 82
1.20 The summation convention......Page 85
Summary......Page 86
Problems......Page 90
Hints and answers......Page 101
2.1 Differentiation of vectors......Page 105
2.1.1 Differentiation of composite vector expressions......Page 108
2.1.2 Differential of a vector......Page 109
2.2 Integration of vectors......Page 110
2.3 Vector functions of several arguments......Page 111
2.4 Surfaces......Page 112
2.6 Vector operators......Page 114
2.6.1 Gradient of a scalar field......Page 115
2.6.2 Divergence of a vector field......Page 118
2.6.3 Curl of a vector field......Page 120
2.7 Vector operator formulae......Page 121
2.7.1 Vector operators acting on sums and products......Page 122
2.7.2 Combinations of grad, div and curl......Page 123
2.8.1 Cylindrical polar coordinates......Page 125
2.8.2 Spherical polar coordinates......Page 129
2.9 General curvilinear coordinates......Page 131
2.9.1 Gradient......Page 134
2.9.2 Divergence......Page 135
2.9.4 Curl......Page 136
Summary......Page 137
Problems......Page 139
Hints and answers......Page 144
3.1 Line integrals......Page 146
3.1.1 Evaluating line integrals......Page 147
3.1.2 Physical examples of line integrals......Page 150
3.1.3 Line integrals with respect to a scalar......Page 151
3.2 Connectivity of regions......Page 152
3.3 Green's theorem in a plane......Page 153
3.4 Conservative fields and potentials......Page 156
3.5 Surface integrals......Page 159
3.5.1 Evaluating surface integrals......Page 160
3.5.2 Vector areas of surfaces......Page 163
3.6 Volume integrals......Page 165
3.7 Integral forms for grad, div and curl......Page 167
3.8 Divergence theorem and related theorems......Page 171
3.8.1 Green's theorems......Page 172
3.8.2 Other related integral theorems......Page 173
3.8.3 Physical applications of the divergence theorem......Page 174
3.9 Stokes' theorem and related theorems......Page 176
3.9.1 Related integral theorems......Page 177
3.9.2 Physical applications of Stokes' theorem......Page 178
Summary......Page 179
Problems......Page 181
Hints and answers......Page 186
4.1 The Dirichlet conditions......Page 188
4.2 The Fourier coefficients......Page 190
4.3 Symmetry considerations......Page 192
4.4 Discontinuous functions......Page 193
4.5 Non-periodic functions......Page 194
4.6 Integration and differentiation......Page 197
4.7 Complex Fourier series......Page 198
4.8 Parseval's theorem......Page 199
Summary......Page 201
Problems......Page 203
Hints and answers......Page 207
5.1 Fourier transforms......Page 209
5.1.1 The uncertainty principle......Page 211
5.1.2 Fraunhofer diffraction......Page 213
5.2 The Dirac deltaf-unction......Page 215
5.2.1 Relation of the deltaf-function to Fourier transforms......Page 218
5.3 Properties of Fourier transforms......Page 220
5.3.1 Odd and even functions......Page 221
5.3.2 Convolution and deconvolution......Page 222
5.3.3 Parseval's theorem......Page 226
5.3.4 Fourier transforms in higher dimensions......Page 227
5.4 Laplace transforms......Page 228
5.4.1 Laplace transforms of derivatives and integrals......Page 231
5.4.2 Other properties of Laplace transforms......Page 232
5.5 Concluding remarks......Page 235
Summary......Page 236
Problems......Page 237
Hints and answers......Page 244
6 Higher-order ordinary differential equations......Page 246
6.1.1 General form of solution......Page 247
6.1.2 Linear equations......Page 248
6.2.1 Finding the complementary function yc(x)......Page 251
6.2.2 Finding the particular integral yp(x)......Page 253
6.2.3 Constructing the general solution yc(x)+yp(x)......Page 254
6.3 Linear recurrence relations......Page 255
6.3.1 First-order recurrence relations......Page 256
6.3.2 Second-order recurrence relations......Page 258
6.4 Laplace transform method......Page 260
6.5.1 The Legendre and Euler linear equations......Page 262
6.5.2 Exact equations......Page 264
6.5.3 Partially known complementary function......Page 266
6.5.4 Variation of parameters......Page 268
6.5.5 Green's functions......Page 270
6.6.1 Dependent variable absent......Page 276
6.6.2 Independent variable absent......Page 277
6.6.3 Equations homogeneous in x or y alone......Page 278
Summary......Page 280
Problems......Page 282
Hints and answers......Page 289
7.1 Second-order linear ordinary differential equations......Page 291
7.2 Ordinary and singular points of an ODE......Page 293
7.3 Series solutions about an ordinary point......Page 295
7.4 Series solutions about a regular singular point......Page 298
7.4.1 Distinct roots not differing by an integer......Page 300
7.4.3 Distinct roots differing by an integer......Page 302
7.5.1 The Wronskian method......Page 304
7.5.2 The derivative method......Page 305
7.5.3 Series form of the second solution......Page 307
7.6 Polynomial solutions......Page 308
Summary......Page 310
Problems......Page 311
Hints and answers......Page 315
8 Eigenfunction methods for differential equations......Page 316
8.1 Sets of functions......Page 318
8.2 Adjoint, self-adjoint and Hermitian operators......Page 321
8.3.2 Orthogonality and normalization of the eigenfunctions......Page 323
8.3.4 Construction of real eigenfunctions......Page 325
8.4.1 Hermitian nature of the Sturm--Liouville operator......Page 326
8.4.2 Transforming an equation into Sturm--Liouville form......Page 327
8.5 Superposition of eigenfunctions: Green's functions......Page 330
Summary......Page 333
Problems......Page 334
Hints and answers......Page 338
9.1 Legendre functions......Page 340
9.1.1 Legendre functions for integer l......Page 341
9.1.2 Properties of Legendre polynomials......Page 344
9.2 Associated Legendre functions......Page 351
9.2.1 Associated Legendre functions for integer l......Page 352
Mutual orthogonality......Page 354
Recurrence relations......Page 356
9.3 Spherical harmonics......Page 357
9.4 Chebyshev functions......Page 359
9.4.1 Properties of Chebyshev polynomials......Page 362
Recurrence relations......Page 364
9.5 Bessel functions......Page 365
9.5.1 Bessel functions for non-integer nu......Page 366
9.5.2 Bessel functions for integer nu......Page 368
Mutual orthogonality......Page 371
Recurrence relations......Page 374
Generating function......Page 375
Integral representations......Page 376
9.6 Spherical Bessel functions......Page 378
9.7 Laguerre functions......Page 379
Mutual orthogonality......Page 381
Generating function......Page 382
Recurrence relations......Page 383
9.8 Associated Laguerre functions......Page 384
Mutual orthogonality......Page 385
Recurrence relations......Page 386
9.9 Hermite functions......Page 387
Rodrigues’ formula......Page 388
Mutual orthogonality......Page 389
Generating function......Page 390
9.10.1 The gamma function......Page 391
9.10.2 The incomplete gamma function......Page 393
9.10.3 The error function......Page 394
Summary......Page 395
Problems......Page 398
Hints and answers......Page 403
10.1 Important partial differential equations......Page 405
10.1.1 The wave equation......Page 406
10.1.2 The diffusion equation......Page 407
10.1.5 Schrodinger's equation......Page 409
10.2 General form of solution......Page 410
10.3.1 First-order equations......Page 411
10.3.2 Inhomogeneous equations and problems......Page 415
10.3.3 Second-order equations......Page 417
10.4 The wave equation......Page 423
10.5 The diffusion equation......Page 426
10.6 Boundary conditions and the uniqueness of solutions......Page 429
10.6.1 Uniqueness of solutions......Page 430
Summary......Page 431
Problems......Page 432
Hints and answers......Page 437
11.1 Separation of variables: the general method......Page 439
11.2 Superposition of separated solutions......Page 443
Laplace’s equation in plane polars......Page 451
Laplace’s equation in spherical polars......Page 457
11.3.2 Other equations in polar coordinates......Page 463
Helmholtz’s equation in plane polars......Page 464
Helmholtz’s equation in cylindrical polars......Page 465
11.3.3 Solution by expansion......Page 467
11.3.4 Separation of variables for inhomogeneous equations......Page 470
11.4 Integral transform methods......Page 473
11.5 Inhomogeneous problems -- Green's functions......Page 478
11.5.1 Similarities to Green's functions for ODEs......Page 479
11.5.2 General boundary-value problems......Page 480
11.5.3 Dirichlet problems......Page 482
11.5.4 Neumann problems......Page 491
Summary......Page 494
Problems......Page 497
Hints and answers......Page 504
12 Calculus of variations......Page 506
12.1 The Euler--Lagrange equation......Page 507
12.2.1 F does not contain y explicitly......Page 508
12.2.2 F does not contain x explicitly......Page 510
12.3 Some extensions......Page 512
12.3.3 Higher-order derivatives......Page 513
12.4 Constrained variation......Page 514
12.5.1 Fermat's principle in optics......Page 516
12.5.2 Hamilton's principle in mechanics......Page 517
12.6 General eigenvalue problems......Page 519
12.7 Estimation of eigenvalues and eigenfunctions......Page 521
12.8 Adjustment of parameters......Page 524
Summary......Page 525
Problems......Page 527
Hints and answers......Page 532
13.1 Obtaining an integral equationfrom a differential equation......Page 534
13.2 Types of integral equation......Page 535
13.3 Operator notation and the existence of solutions......Page 536
13.4.1 Separable kernels......Page 537
13.4.2 Integral transform methods......Page 539
13.4.3 Differentiation......Page 543
13.5 Neumann series......Page 544
13.6 Fredholm theory......Page 546
13.7 Schmidt--Hilbert theory......Page 547
Summary......Page 550
Problems......Page 552
Hints and answers......Page 556
14 Complex variables......Page 558
14.1 Functions of a complex variable......Page 559
14.2 The Cauchy--Riemann relations......Page 561
14.3 Power series in a complex variable......Page 565
14.4 Some elementary functions......Page 567
14.5 Multivalued functions and branch cuts......Page 569
14.6 Singularities and zeros of complex functions......Page 571
14.7 Conformal transformations......Page 574
14.8 Complex integrals......Page 577
14.9 Cauchy's theorem......Page 581
14.10 Cauchy's integral formula......Page 584
14.11 Taylor and Laurent series......Page 586
14.12 Residue theorem......Page 591
Summary......Page 594
Problems......Page 596
Hints and answers......Page 598
15.1 Complex potentials......Page 600
15.2 Applications of conformal transformations......Page 605
15.3 Definite integrals using contour integration......Page 608
15.3.1 Integrals of sinusoidal functions......Page 609
15.3.2 Some infinite integrals......Page 610
15.3.3 Integrals of multivalued functions......Page 614
15.4 Summation of series......Page 615
15.5 Inverse Laplace transform......Page 617
15.6 Some more advanced applications......Page 620
Summary......Page 623
Problems......Page 624
Hints and answers......Page 628
16.1 Venn diagrams......Page 630
16.2 Probability......Page 635
16.2.1 Axioms and theorems......Page 636
16.2.2 Conditional probability......Page 640
16.2.3 Bayes' theorem......Page 643
16.3.1 Permutations......Page 645
16.3.2 Combinations......Page 647
16.4.1 Discrete random variables......Page 651
16.4.2 Continuous random variables......Page 653
16.4.3 Sets of random variables......Page 654
16.5.1 Mean......Page 656
16.5.2 Mode and median......Page 657
16.5.3 Variance and standard deviation......Page 658
16.5.4 Moments......Page 659
16.6.1 Continuous random variables......Page 660
16.6.2 Functions of several random variables......Page 661
16.6.3 Expectation values and variances......Page 662
16.7.1 Probability generating functions......Page 664
Sums of random variables......Page 666
Variable-length sums of random variables......Page 668
16.7.2 Moment generating functions......Page 669
Scaling and shifting......Page 670
Uniqueness......Page 671
16.8.1 The binomial distribution......Page 672
The moment generating function for the binomial distribution......Page 674
16.8.2 The multinomial distribution......Page 675
16.8.3 The geometric and negative binomial distributions......Page 677
16.8.4 The hypergeometric distribution......Page 678
16.8.5 The Poisson distribution......Page 679
The Poisson approximation to the binomial distribution......Page 682
Multiple Poisson distributions......Page 683
16.9 Important continuous distributions......Page 684
16.9.2 The Gaussian distribution......Page 685
Gaussian approximation to the binomial distribution......Page 691
Gaussian approximation to the Poisson distribution......Page 693
Multiple Gaussian distributions......Page 694
16.9.3 The exponential and gamma distributions......Page 696
16.9.4 The chi-squared distribution......Page 697
16.9.5 The Cauchy and Breit--Wigner distributions......Page 698
16.10 The central limit theorem......Page 699
16.11.1 Discrete bivariate distributions......Page 701
16.11.2 Continuous bivariate distributions......Page 702
16.12 Properties of joint distributions......Page 703
16.12.3 Covariance and correlation......Page 704
Summary......Page 709
Problems......Page 713
Hints and answers......Page 721
17.1 Experiments, samples and populations......Page 723
17.2.1 Averages......Page 724
17.2.2 Variance and standard deviation......Page 726
17.2.4 Covariance and correlation......Page 728
17.3 Estimators and sampling distributions......Page 731
Bias......Page 732
Efficiency......Page 733
17.3.2 Standard errors on estimators......Page 734
17.3.3 Confidence limits on estimators......Page 736
17.3.4 Confidence limits for a Gaussian sampling distribution......Page 738
17.4.1 Population mean mu......Page 739
17.4.2 Population variance a2......Page 740
17.4.3 Population standard deviation a......Page 742
17.4.4 Population moments ur......Page 743
17.4.5 Population covariance Cov[x,y] and correlation Corr[x,y]......Page 744
17.4.6 A worked example......Page 746
17.5.1 The method of least squares......Page 748
17.5.2 Linear least squares......Page 750
17.6.1 Simple and composite hypotheses......Page 753
17.6.2 Statistical tests......Page 754
17.6.3 The generalized likelihood-ratio test......Page 755
17.6.4 Student's t-test......Page 758
17.6.5 Fisher's F-test......Page 764
17.6.6 Goodness of fit in least-squares problems......Page 769
17.6.7 Elementary contingency analysis......Page 771
Summary......Page 773
Problems......Page 777
Hints and answers......Page 782
A.1 Arithmetic and geometry......Page 784
A.2 Preliminary algebra......Page 786
A.3 Differential calculus......Page 788
A.4 Integral calculus......Page 789
A.5 Complex numbers and hyperbolic functions......Page 791
A.6 Series and limits......Page 792
A.7 Partial differentiation......Page 795
A.8 Multiple integrals......Page 796
A.9 Vector algebra......Page 797
A.10 First-order ordinary differential equations......Page 799
Appendix B: Inner products......Page 800
Appendix C: Inequalities in linear vector spaces......Page 802
Appendix D: Summation convention......Page 804
Appendix E: The Kronecker delta and Levi--Civita symbols......Page 807
Appendix F: Gram-Schmidt orthogonalization......Page 811
Appendix G: Linear least squares......Page 813
1. Matrices and vector spaces......Page 815
2. Vector calculus......Page 818
4. Fourier series......Page 819
6. Higher-order ordinary differential equations......Page 820
7. Series solutions of ordinary differential equations......Page 821
9. Special functions......Page 822
10. Partial differential equations......Page 823
14. Complex variables......Page 824
16. Probability......Page 825
17. Statistics......Page 826
Index......Page 828