Stone M., Goldbart P. Mathematics for Physics
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(Оффтоп)
Preface page xi
Acknowledgments xiii
1 Calculus of variations 1
1.1 What is it good for? 1
1.2 Functionals 1
1.3 Lagrangian mechanics 10
1.4 Variable endpoints 27
1.5 Lagrange multipliers 32
1.6 Maximum or minimum? 36
1.7 Further exercises and problems 38
2 Function spaces 50
2.1 Motivation 50
2.2 Norms and inner products 51
2.3 Linear operators and distributions 66
2.4 Further exercises and problems 76
3 Linear ordinary differential equations 86
3.1 Existence and uniqueness of solutions 86
3.2 Normal form 93
3.3 Inhomogeneous equations 94
3.4 Singular points 97
3.5 Further exercises and problems 98
4 Linear differential operators 101
4.1 Formal vs. concrete operators 101
4.2 The adjoint operator 104
4.3 Completeness of eigenfunctions 117
4.4 Further exercises and problems 132
5 Green functions 140
5.1 Inhomogeneous linear equations 140
5.2 Constructing Green functions 141
viiviii Contents
5.3 Applications of Lagrange’s identity 150
5.4 Eigenfunction expansions 153
5.5 Analytic properties of Green functions 155
5.6 Locality and the Gelfand–Dikii equation 165
5.7 Further exercises and problems 167
6 Partial differential equations 174
6.1 Classification of PDEs 174
6.2 Cauchy data 176
6.3 Wave equation 181
6.4 Heat equation 196
6.5 Potential theory 201
6.6 Further exercises and problems 224
7 The mathematics of real waves 231
7.1 Dispersive waves 231
7.2 Making waves 242
7.3 Nonlinear waves 246
7.4 Solitons 255
7.5 Further exercises and problems 260
8 Special functions 264
8.1 Curvilinear coordinates 264
8.2 Spherical harmonics 270
8.3 Bessel functions 278
8.4 Singular endpoints 298
8.5 Further exercises and problems 305
9 Integral equations 311
9.1 Illustrations 311
9.2 Classification of integral equations 312
9.3 Integral transforms 313
9.4 Separable kernels 321
9.5 Singular integral equations 323
9.6 Wiener–Hopf equations I 327
9.7 Some functional analysis 332
9.8 Series solutions 338
9.9 Further exercises and problems 342
10 Vectors and tensors 347
10.1 Covariant and contravariant vectors 347
10.2 Tensors 350
10.3 Cartesian tensors 362
10.4 Further exercises and problems 372Contents ix
11 Differential calculus on manifolds 376
11.1 Vector and covector fields 376
11.2 Differentiating tensors 381
11.3 Exterior calculus 389
11.4 Physical applications 395
11.5 Covariant derivatives 403
11.6 Further exercises and problems 409
12 Integration on manifolds 414
12.1 Basic notions 414
12.2 Integrating p-forms 417
12.3 Stokes’ theorem 422
12.4 Applications 424
12.5 Further exercises and problems 440
13 An introduction to differential topology 449
13.1 Homeomorphism and diffeomorphism 449
13.2 Cohomology 450
13.3 Homology 455
13.4 De Rham’s theorem 469
13.5 Poincaré duality 473
13.6 Characteristic classes 477
13.7 Hodge theory and the Morse index 483
13.8 Further exercises and problems 496
14 Groups and group representations 498
14.1 Basic ideas 498
14.2 Representations 505
14.3 Physics applications 517
14.4 Further exercises and problems 525
15 Lie groups 530
15.1 Matrix groups 530
15.2 Geometry of SU(2) 535
15.3 Lie algebras 555
15.4 Further exercises and problems 572
16 The geometry of fibre bundles 576
16.1 Fibre bundles 576
16.2 Physics examples 577
16.3 Working in the total space 591
17 Complex analysis 606
17.1 Cauchy–Riemann equations 606x Contents
17.2 Complex integration: Cauchy and Stokes 616
17.3 Applications 624
17.4 Applications of Cauchy’s theorem 630
17.5 Meromorphic functions and the winding number 644
17.6 Analytic functions and topology 647
17.7 Further exercises and problems 661
18 Applications of complex variables 666
18.1 Contour integration technology 666
18.2 The Schwarz reflection principle 676
18.3 Partial-fraction and product expansions 687
18.4 Wiener–Hopf equations II 692
18.5 Further exercises and problems 701
19 Special functions and complex variables 706
19.1 The Gamma function 706
19.2 Linear differential equations 711
19.3 Solving ODEs via contour integrals 718
19.4 Asymptotic expansions 725
19.5 Elliptic functions 735
19.6 Further exercises and problems 741
A Linear algebra review 744
A.1 Vector space 744
A.2 Linear maps 746
A.3 Inner-product spaces 749
A.4 Sums and differences of vector spaces 754
A.5 Inhomogeneous linear equations 757
A.6 Determinants 759
A.7 Diagonalization and canonical forms 766
B Fourier series and integrals 779
B.1 Fourier series 779
B.2 Fourier integral transforms 783
B.3 Convolution 786
B.4 The Poisson summation formula 792
References 797
Index 799
