Potential Theory in Gravity and Magnetic Applications
Table of Contents
Introduction
1. The Potential
- Potential Fields
- Fields
- Points, Boundaries, and Regions
- Energy, Work, and the Potential
- Harmonic Functions
- Laplace's Equation
- An Example from Steady-State Heat Flow
- Complex Harmonic Functions
- Problem Set
2. Consequences of the Potential
- Green's Identities
- Green's First Identity
- Green's Second Identity
- Green's Third Identity
- Gauss' Theorem of the Arithmetic Mean
- Helmholtz Theorem
- Proof of the Helmholtz Theorem
- Consequences of the Helmholtz Theorem
- Example
- Green's Functions
- Analogy with Linear Systems
- Green's Functions and Laplace's Equation
- Problem Set
3. Newtonian Potential
- Gravitational Attraction and Potential
- The Potential of Distributions of Mass
- Example: A Spherical Shell
- Example: Solid Sphere
- Example: Straight Wire of Finite Length
- Potential of Two-Dimensional Distributions
- Potential of an Infinite Wire
- General Two-Dimensional Distributions
- Gauss' Law for Gravity Fields
- Green's Equivalent Layer
- Problem Set
4. Magnetic Potential
- Magnetic Induction
- Gauss' Law for Magnetic Fields
- The Vector and Scalar Potential
- Dipole Moment and Potential
- First Derivation: Two Current Loops
- Second Derivation: Two monopoles
- Dipole Field
- Problem Set
5. Magnetization
- Distributions of Magnetization
- Magnetic Field Intensity
- Magnetic Permeability and Susceptibility
- Poisson's Relation
- Example: A Sphere
- Example: Infinite Slab
- Example: Horizontal Cylinder
- Two-dimensional Distributions of Magnetization
- Annihilators
- Problem Set
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6. Spherical Harmonic Analysis
- Introduction
- Zonal Harmonics
- Surface Harmonics
- Normalized Functions
- Tesseral and Sectoral Surface Harmonics
- Application to Laplace's Equation
- Homogeneous Functions and Euler's Equation
- Point Source Away from Origin
- General Spherical Surface Harmonic Functions
- Problem Set
7. Regional Gravity Fields
- Introduction
- The "Normal" Earth
- Gravity Anomalies
- Free-Air Correction
- Tide Correction
- Eötvös Correction
- Bouguer Correction
- Isostatic Residual
- An Example
- Problem Set
8. The Geomagnetic Field
- Parts of Internal and External Origin
- Description of the Geomagnetic Field
- The Elements of the Geomagnetic Field
- The International Geomagnetic Reference Field
- The Dipole Field
- The Nondipole Field
- Secular Variation
- Crustal Magnetic Anomalies
- Problem Set
9. Forward Method
- Methods Compared
- Gravity Models
- Three-Dimensional Examples
- Two-Dimensional Examples
- Magnetic Models
- A Choice of Models
- Three-Dimensional Examples
- Two-Dimensional Example
- Problem Set
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10. Inverse Method
- Introduction
- Linear Inverse Problem
- Magnetization of a Layer
- Determination of Magnetization Direction
- Nonlinear Inverse Problem
- Shape of Source
- Depth to Source
- Ideal Bodies
- Problem Set
11. Fourier-Domain Modeling
- Notation and Review
- Fourier Transform
- Properties of Fourier Transforms
- Random Functions
- Generalized Functions
- Convolution
- Discrete Fourier Transform
- Some Simple Anomalies
- Three-Dimensional Sources
- Two-Dimensional Sources
- Earth Filters
- Parker's Method
- General Sources
- Depth and Shape of Source
- Statistical Models
- Depth to Bottom
- Problem Set
12. Transformations
- Upward Continuation
- Level Surface to Level Surface
- Uneven Surfaces
- Directional Derivatives
- Phase Transformations
- Reduction to the Pole
- Calculation of Vector Components
- Pseudogravity Transformation
- Pseudomagnetic Calculation
- Horizontal Gradients and Boundary Analysis
- Analytic Signal
- Hilbert Transforms
- Application to Potential Fields
- Problem Set
Appendices
- Review of Vector Calculus
- Subroutines
- Review of Sampling Theory
- Conversion of Units
Bibliography
Index
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Return to Potential Theory
Cambridge University Press (United Kingdom)
Cambridge University Press (North America)