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Introduction | |

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Power series | |

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The Taylor series | |

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Growth of the Earth by cosmic dust | |

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The bouncing ball | |

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Reflection and transmission by a stack of layers | |

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Spherical and cylindrical coordinates | |

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Introducing spherical coordinates | |

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Changing coordinate systems | |

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The acceleration in spherical coordinates | |

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Volume integration in spherical coordinates | |

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Cylindrical coordinates | |

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The gradient | |

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Properties of the gradient vector | |

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The pressure force | |

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Differentiation and integration | |

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Newton's law from energy conservation | |

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Total and partial time derivatives | |

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The gradient in spherical coordinates | |

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The divergence of a vector field | |

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The flux of a vector field | |

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Introduction of the divergence | |

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Sources and sinks | |

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The divergence in cylindrical coordinates | |

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Is life possible in a five-dimensional world? | |

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The curl of a vector field | |

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Introduction of the curl | |

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What is the curl of the vector field? | |

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The first source of vorticity: rigid rotation | |

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The second source of vorticity: shear | |

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The magnetic field induced by a straight current | |

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Spherical coordinates and cylindrical coordinates | |

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The theorem of Gauss | |

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Statement of Gauss's law | |

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The gravitational field of a spherically symmetric mass | |

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A representation theorem for acoustic waves | |

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Flowing probability | |

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The theorem of Stokes | |

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Statement of Stokes's law | |

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Stokes's theorem from the theorem of Gauss | |

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The magnetic field of a current in a straight wire | |

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Magnetic induction and Lenz's law | |

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The Aharonov--Bohm effect | |

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Wingtips vortices | |

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The Laplacian | |

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The curvature of a function | |

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The shortest distance between two points | |

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The shape of a soap film | |

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Sources of curvature | |

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The instability of matter | |

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Where does lightning start? | |

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The Laplacian in spherical and cylindrical coordinates | |

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Averaging integrals for harmonic functions | |

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Conservation laws | |

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The general form of conservation laws | |

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The continuity equation | |

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Conservation of momentum and energy | |

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The heat equation | |

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The explosion of a nuclear bomb | |

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Viscosity and the Navier--Stokes equation | |

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Quantum mechanics and hydrodynamics | |

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Scale analysis | |

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The vortex in a bathtub | |

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Three ways to estimate a derivative | |

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The advective terms in the equation of motion | |

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Geometric ray theory | |

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Is the Earth's mantle convecting? | |

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Making an equation dimensionless | |

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Linear algebra | |

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Projections and the completeness relation | |

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A projection on vectors that are not orthogonal | |

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The Coriolis force and centrifugal force | |

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The eigenvalue decomposition of a square matrix | |

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Computing a function of a matrix | |

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The normal modes of a vibrating system | |

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Singular value decomposition | |

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The Householder transformation | |

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The Dirac delta function | |

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Introduction of the delta function | |

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Properties of the delta function | |

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The delta function of a function | |

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The delta function in more dimensions | |

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The delta function on the sphere | |

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The self energy of the electron | |

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Fourier analysis | |

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The real Fourier series on a finite interval | |

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The complex Fourier series on a finite interval | |

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The Fourier transform on an infinite interval | |

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The Fourier transform and the delta function | |

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Changing the sign and scale factor | |

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The convolution and correlation of two signals | |

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Linear filters and the convolution theorem | |

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The dereverberation filter | |

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Design of frequency filters | |

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Linear filters and linear algebra | |

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Analytic functions | |

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The theorem of Cauchy--Riemann | |

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The electric potential | |

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Fluid flow and analytic functions | |

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Complex integration | |

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Nonanalytic functions | |

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The residue theorem | |

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Solving integrals without knowing the primitive function | |

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Response of a particle in syrup | |

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Green's functions: principles | |

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The girl on a swing | |

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You have seen Green's functions before! | |

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The Green's function as impulse response | |

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The Green's function for a general problem | |

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Radiogenic heating and the Earth's temperature | |

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Nonlinear systems and Green's functions | |

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Green's functions: examples | |

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The heat equation in N dimensions | |

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The Schrodinger equation with an impulsive source | |

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The Helmholtz equation in one, two and three dimensions | |

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The wave equation in one, two and three dimensions | |

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Normal modes | |

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The normal modes of a string | |

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The normal modes of drum | |

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The normal modes of a sphere | |

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Normal modes and orthogonality relations | |

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Bessel functions behave as decaying cosines | |

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Legendre functions behave as decaying cosines | |

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Normal modes and the Green's function | |

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Guided waves in a low-velocity channel | |

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Leaky modes | |

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Radiation damping | |

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Potential theory | |

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The Green's function of the gravitational potential | |

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Upward continuation in a flat geometry | |

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Upward continuation in a flat geometry in three dimensions | |

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The gravity field of the Earth | |

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Dipoles, quadrupoles and general relativity | |

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The multipole expansion | |

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The quadrupole field of the Earth | |

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The fifth force | |

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Cartesian tensors | |

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Coordinate transforms | |

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Unitary matrices | |

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Shear or dilatation? | |

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The summation convention | |

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Matrices and coordinate transforms | |

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Definition of a tensor | |

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Not every vector is a tensor | |

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The products of tensors | |

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Deformation and rotation again | |

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The stress tensor | |

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Why pressure in a fluid is isotropic | |

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Special relativity | |

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Perturbation theory | |

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Regular perturbation theory | |

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The Born approximation | |

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Linear travel time tomography | |

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Limits on perturbation theory | |

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The WKB approximation | |

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The need for consistency | |

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Singular perturbation theory | |

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Epilogue, on power and knowledge | |

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References | |

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Index | |