Preface ........................................................ xi
Acknowledgments .............................................. xiii
1 Mathematical background ...................................... 1
1.1 Cartesian and spherical coordinates ..................... 1
1.2 Complex numbers ......................................... 1
1.3 Vector relationships .................................... 4
1.4 Matrices and tensors .................................... 8
1.5 Conservative force, field, and potential ............... 17
1.6 The divergence theorem (Gauss's theorem) ............... 18
1.7 The curl theorem (Stokes'theorem) ...................... 20
1.8 Poisson's equation ..................................... 23
1.9 Laplace's equation ..................................... 26
1.10 Power series ........................................... 28
1.11 Leibniz's rule ......................................... 32
1.12 Legendre polynomials ................................... 32
1.13 The Legendre differential equation ..................... 34
1.14 Rodrigues' formula ..................................... 41
1.15 Associated Legendre polynomials ........................ 43
1.16 Spherical harmonic functions ........................... 49
1.17 Fourier series, Fourier integrals, and Fourier
transforms ............................................. 52
Further reading ............................................. 58
2 Gravitation ................................................. 59
2.1 Gravitational acceleration and potential ............... 59
2.2 Kepler's laws of planetary motion ...................... 60
2.3 Gravitational acceleration and the potential of
a solid sphere ......................................... 66
2.4 Laplace's equation in spherical polar coordinates ...... 69
2.5 MacCullagh's formula for the gravitational potential ... 74
Further reading ............................................. 85
3 Gravity ..................................................... 86
3.1 The ellipticity of the Earth's figure .................. 86
3.2 The geopotential ....................................... 88
3.3 The equipotential surface of gravity ................... 91
3.4 Gravity on the reference spheroid ...................... 96
3.5 Geocentric and geographic latitude .................... 102
3.6 The geoid ............................................. 106
Further reading ............................................ 115
4 The tides .................................................. 116
4.1 Origin of the lunar tide-raising forces ............... 116
4.2 Tidal potential of the Moon ........................... 119
4.3 Love's numbers and the tidal deformation .............. 124
4.4 Tidal friction and deceleration of terrestrial and
lunar rotations ....................................... 130
Further reading ............................................ 136
5 Earth's rotation ........................................... 137
5.1 Motion in a rotating coordinate system ................ 138
5.2 The Coriolis and Eötvös effects ....................... 140
5.3 Precession and forced nutation of Earth's rotation
axis .................................................. 142
5.4 The free, Eulerian nutation of a rigid Earth .......... 155
5.5 The Chandler wobble ................................... 157
Further reading ............................................ 169
6 Earth's heat ............................................... 170
6.1 Energy and entropy .................................... 171
6.2 Thermodynamic potentials and Maxwell's relations ...... 172
6.3 The melting-temperature gradient in the core ......... 176
6.4 The adiabatic temperature gradient in the core ........ 178
6.5 The Grьneisen parameter ............................... 179
6.6 Heat flow ............................................. 182
Further reading ............................................ 197
7 Geomagnetism ............................................... 198
7.1 The dipole magnetic field and potential ............... 198
7.2 Potential of the geomagnetic field .................... 200
7.3 The Earth's dipole magnetic field ..................... 205
7.4 Secular variation ..................................... 213
7.5 Power spectrum of the internal field .................. 214
7.6 The origin of the internal field ...................... 217
Further reading ............................................ 225
8 Foundations of seismology .................................. 227
8.1 Elastic deformation .................................. 227
8.2 Stress ................................................ 228
8.3 Strain ................................................ 233
8.4 Perfectly elastic stress-strain relationships ......... 239
8.5 The seismic wave equation ............................. 244
8.6 Solutions of the wave equation ........................ 252
8.7 Three-dimensional propagation of plane P-and S-waves .. 254
Further reading ........................................... 258
Appendix A Magnetic poles, the dipole field, and current
loops ...................................................... 259
Appendix В Maxwell s equations of electromagnetism ........... 265
References .................................................... 276
Index ......................................................... 278
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