Geomathematically Oriented Potential Theory

Geomathematically Oriented Potential Theory

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ISBN 9781439895429
Cat# K14227
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Features

  • Presents parallel discussions of three-dimensional Euclidean space and spherical potential theory
  • Describes extensive applications to geoscientific problems, including modeling from satellite data
  • Provides a balanced combination of rigorous mathematics with the geosciences
  • Includes new space-localizing methods for the multiscale analysis of the gravitational and geomagnetic field

Summary

As the Earth`s surface deviates from its spherical shape by less than 0.4 percent of its radius and today’s satellite missions collect their gravitational and magnetic data on nearly spherical orbits, sphere-oriented mathematical methods and tools play important roles in studying the Earth’s gravitational and magnetic field.

Geomathematically Oriented Potential Theory presents the principles of space and surface potential theory involving Euclidean and spherical concepts. The authors offer new insight on how to mathematically handle gravitation and geomagnetism for the relevant observables and how to solve the resulting potential problems in a systematic, mathematically rigorous framework.

The book begins with notational material and the necessary mathematical background. The authors then build the foundation of potential theory in three-dimensional Euclidean space and its application to gravitation and geomagnetism. They also discuss surface potential theory on the unit sphere along with corresponding applications.

Focusing on the state of the art, this book breaks new geomathematical grounds in gravitation and geomagnetism. It explores modern sphere-oriented potential theoretic methods as well as classical space potential theory.

Table of Contents

PRELIMINARIES
Three-Dimensional Euclidean Space R
3
Basic Notation
Integral Theorems

Two-Dimensional Sphere Ω
Basic Notation
Integral Theorems
(Scalar) Spherical Harmonics
(Scalar) Circular Harmonics
Vector Spherical Harmonics
Tensor Spherical Harmonics

POTENTIAL THEORY IN THE EUCLIDEAN SPACE R3
Basic Concepts
Background Material
Volume Potentials
Surface Potentials
Boundary-Value Problems
Locally and Globally Uniform Approximation

Gravitation
Oblique Derivative Problem
Satellite Problems
Gravimetry Problem

Geomagnetism
Geomagnetic Background
Mie and Helmholtz Decomposition
Gauss Representation and Uniqueness
Separation of Sources
Ionospheric Current Systems

POTENTIAL THEORY ON THE UNIT SPHERE Ω
Basic Concepts
Background Material
Surface Potentials
Curve Potentials
Boundary-Value Problems
Differential Equations for Surface Gradient and Surface Curl Gradient
Locally and Globally Uniform Approximation

Gravitation
Disturbing Potential
Linear Regularization Method
Multiscale Solution

Geomagnetics
Mie and Helmholtz Decomposition
Higher-Order Regularization Methods
Separation of Sources
Ionospheric Current Systems

Bibliography

Index

Exercises appear at the end of each chapter.

Author Bio(s)

Willi Freeden is a professor in the Geomathematics Group at the University of Kaiserslautern. Dr. Freeden is an editorial board member of seven international journals and was previously the editor-in-chief of the International Journal on Geomathematics. His research interests include special functions of mathematical (geo)physics, PDEs, constructive approximation, integral transforms, numerical methods, the use of mathematics in industry, and inverse problems in geophysics, geodesy, and satellite technology.

Christian Gerhards is a visiting postdoc researcher in the Department of Mathematics and Statistics at the University of New South Wales.

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