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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