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ОбложкаSnieder R. A guided tour of mathematical methods for the physical sciences. - 2nd ed. - Cambridge; New York: Cambridge University Press, 2004 (2009). - xiv, 507 p.: ill. - Ref.: p.494-499. - Ind.: p.500-507. - ISBN 978-0-521-54261-6
Шифр: (И/В31-S69) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
 
Preface to Second Edition .................................... xiii
Acknowledgements .............................................. xiv

1  Introduction ................................................. 1

2  Dimensional analysis ......................................... 3
   2.1  Two rules for physical analysis ......................... 3
   2.2  A trick for finding mistakes ............................ 6
   2.3  Buckingham pi theorem ................................... 7
   2.4  Lift of a wing ......................................... 11
   2.5  Scaling relations ...................................... 12
   2.6  Dependence of pipe flow on the radius of the pipe ...... 13
   
3  Power series ................................................ 16
   3.1  Taylor series .......................................... 16
   3.2  Growth of the Earth by cosmic dust ..................... 22
   3.3  Bouncing ball .......................................... 24
   3.4  Reflection and transmission by a stack of layers ....... 27
   
4  Spherical and cylindrical coordinates ....................... 31
   4.1  Introducing spherical coordinates ...................... 31
   4.2  Changing coordinate systems ............................ 35
   4.3  Acceleration in spherical coordinates .................. 37
   4.4  Volume integration in spherical coordinates ............ 40
   4.5  Cylindrical coordinates ................................ 43
   
5  Gradient .................................................... 46
   5.1  Properties of the gradient vector ...................... 46
   5.2  Pressure force ......................................... 50
   5.3  Differentiation and integration ........................ 53
   5.4  Newton's law from energy conservation .................. 55
   5.5  Total and partial time derivatives ..................... 57
   5.6  Gradient in spherical coordinates ...................... 61
   
6  Divergence of a vector field ................................ 64
   6.1  Flux of a vector field ................................. 64
   6.2  Introduction of the divergence ......................... 66
   6.3  Sources and sinks ...................................... 69
   6.4  Divergence in cylindrical coordinates .................. 71
   6.5  Is life possible in a five-dimensional world? .......... 73
   
7  Curl of a vector field ...................................... 78
   7.1  Introduction of the curl ............................... 78
   7.2  What is the curl of the vector field? .................. 80
   7.3  First source of vorticity: rigid rotation .............. 81
   7.4  Second source of vorticity: shear ...................... 83
   7.5  Magnetic field induced by a straight current ........... 85
   7.6  Spherical coordinates and cylindrical coordinates ...... 86
   
8  Theorem of Gauss ............................................ 88
   8.1  Statement of Gauss's law ............................... 88
   8.2  Gravitational field of a spherically symmetric mass .... 89
   8.3  Representation theorem for acoustic waves .............. 91
   8.4  Flowing probability .................................... 93
   
9  Theorem of Stokes ........................................... 97
   9.1  Statement of Stokes's law .............................. 97
   9.2  Stokes's theorem from the theorem of Gauss ............ 100
   9.3  Magnetic field of a current in a straight wire ........ 102
   9.4  Magnetic induction and Lenz's law ..................... 103
   9.5  Aharonov-Bohm effect .................................. 104
   9.6  Wingtips vortices ..................................... 108
   
10 Laplacian .................................................. 113
   10.1 Curvature of a function ............................... 113
   10.2 Shortest distance between two points .................. 117
   10.3 Shape of a soap film .................................. 120
   10.4 Sources of curvature .................................. 124
   10.5 Instability of matter ................................. 126
   10.6 Where does lightning start? ........................... 128
   10.7 Laplacian in spherical and cylindrical coordinates .... 129
   10.8 Averaging integrals for harmonic functions ............ 130
   
11 Conservation laws .......................................... 133
   11.1 General form of conservation laws ..................... 133
   11.2 Continuity equation ................................... 135
   11.3 Conservation of momentum and energy ................... 136
   11.4 Heat equation ......................................... 140
   11.5 Explosion of a nuclear bomb ........................... 145
   11.6 Viscosity and the Navier-Stokes equation .............. 147
   11.7 Quantum mechanics and hydrodynamics ................... 150
   
12 Scale analysis ............................................. 153
   12.1 Vortex in a bathtub ................................... 154
   12.2 Three ways to estimate a derivative ................... 156
   12.3 Advective terms in the equation of motion ............. 159
   12.4 Geometric ray theory .................................. 162
   12.5 Is the Earth's mantle convecting? ..................... 167
   12.6 Making an equation dimensionless ...................... 169
   
13 Linear algebra ............................................. 173
   13.1 Projections and the completeness relation ............. 173
   13.2 Projection on vectors that are not orthogonal ......... 177
   13.3 Coriolis force and centrifugal force .................. 179
   13.4 Eigenvalue decomposition of a square matrix ........... 184
   13.5 Computing a function of a matrix ...................... 187
   13.6 Normal modes of a vibrating system .................... 189
   13.7 Singular value decomposition .......................... 192
   13.8 Householder transformation ............................ 197
   
14 Dirac delta function ....................................... 202
   14.1 Introduction of the delta function .................... 202
   14.2 Properties of the delta function ...................... 206
   14.3 Delta function of a function .......................... 208
   14.4 Delta function in more dimensions ..................... 210
   14.5 Delta function on the sphere .......................... 210
   14.6 Self energy of the electron ........................... 212
   
15 Fourier analysis ........................................... 217
   15.1 Real Fourier series on a finite interval .............. 217
   15.2 Complex Fourier series on a finite interval ........... 221
   15.3 Fourier transform on an infinite interval ............. 223
   15.4 Fourier transform and the delta function .............. 224
   15.5 Changing the sign and scale factor .................... 225
   15.6 Convolution and correlation of two signals ............ 228
   15.7 Linear filters and the convolution theorem ............ 231
   15.8 Dereverberation filter ................................ 234
   15.9 Design of frequency filters ........................... 238
   15.10 Linear filters and linear algebra .................... 240
   
16 Analytic functions ......................................... 245
   16.1 Theorem of Cauchy-Riemann ............................. 245
   16.2 Electric potential .................................... 249
   16.3 Fluid flow and analytic functions ..................... 251
   
17 Complex integration ........................................ 254
   17.1 Nonanalytic functions ................................. 254
   17.2 Residue theorem ....................................... 255
   17.3 Solving integrals without knowing the primitive
        function .............................................. 259
   17.4 Response of a particle in syrup ....................... 262
   
18 Green's functions: principles .............................. 267
   18.1 Girl on a swing ....................................... 267
   18.2 You have seen Green's functions before! ............... 272
   18.3 Green's functions as impulse response ................. 273
   18.4 Green's functions for a general problem ............... 276
   18.5 Radiogenic heating and the Earth's temperature ........ 279
   18.6 Nonlinear systems and the Green's functions ........... 284
   
19 Green's functions: examples ................................ 288
   19.1 Heat equation in N dimensions ......................... 288
   19.2 Schrödinger equation with an impulsive source ......... 292
   19.3 Helmholtz equation in one, two, and three dimensions . 296
   19.4 Wave equation in one, two, and three dimensions ....... 302
   19.5 If I can hear you, you can hear me .................... 308
   
20 Normal modes ............................................... 311
   20.1 Normal modes of a string .............................. 312
   20.2 Normal modes of a drum ................................ 314
   20.3 Normal modes of a sphere .............................. 317
   20.4 Normal modes of orthogonality relations ............... 323
   20.5 Bessel functions behave as decaying cosines ........... 327
   20.6 Legendre functions behave as decaying cosines ......... 330
   20.7 Normal modes and the Green's function ................. 334
   20.8 Guided waves in a low-velocity channel ................ 340
   20.9 Leaky modes ........................................... 344
   20.10 Radiation damping .................................... 348
   
21 Potential theory ........................................... 353
   21.1 Green's function of the gravitational potential ....... 354
   21.2 Upward continuation in a flat geometry ................ 356
   21.3 Upward continuation in a flat geometry in three
        dimensions ............................................ 359
   21.4 Gravity field of the Earth ............................ 361
   21.5 Dipoles, quadrupoles, and general relativity .......... 365
   21.6 Multipole expansion ................................... 369
   21.7 Quadrupole field of the Earth ......................... 374
   21.8 Fifth force ........................................... 377
   
22 Cartesian tensors .......................................... 379
   22.1 Coordinate transforms ................................. 379
   22.2 Unitary matrices ...................................... 382
   22.3 Shear or dilatation? .................................. 385
   22.4 Summation convention .................................. 389
   22.5 Matrices and coordinate transforms .................... 391
   22.6 Definition of a tensor ................................ 393
   22.7 Not every vector is a tensor .......................... 396
   22.8 Products of tensors ................................... 398
   22.9 Deformation and rotation again ........................ 401
   22.10 Stress tensor ........................................ 403
   22.11 Why pressure in a fluid is isotropic ................. 406
   22.12 Special relativity ................................... 408
   
23 Perturbation theory ........................................ 412
   23.1 Regular perturbation theory ........................... 413
   23.2 Bom approximation ..................................... 417
   23.3 Linear travel time tomography ......................... 421
   23.4 Limits on perturbation theory ......................... 424
   23.5 WKB approximation ..................................... 427
   23.6 Need for consistency .................................. 431
   23.7 Singular perturbation theory .......................... 433
   
24 Asymptotic evaluation of integrals ......................... 437
   24.1 Simplest tricks ....................................... 437
   24.2 What does n! have to do with e and spn? ............... 441
   24.3 Method of steepest descent ............................ 445
   24.4 Group velocity and the method of stationary phase ..... 450
   24.5 Asymptotic behavior of the Bessel function Jo(x) ...... 453
   24.6 Image source .......................................... 456
   
25 Variational calculus ....................................... 461
   25.1 Designing a can ....................................... 461
   25.2 Why are cans round? ................................... 463
   25.3 Shortest distance between two points .................. 465
   25.4 The great-circle ...................................... 468
   25.5 Euler-Lagrange equation ............................... 472
   25.6 Lagrangian formulation of classical mechanics ......... 476
   25.7 Rays are curves of stationary travel time ............. 478
   25.8 Lagrange multipliers .................................. 481
   25.9 Designing a can with an optimal shape ................. 485
   25.10 The chain line ....................................... 487
   
26 Epilogue, on power and knowledge ........................... 492
   
References .................................................... 494
Index ......................................................... 500

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