eBook

- 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

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.

**PRELIMINARIESThree-Dimensional Euclidean Space R**

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 R ^{3}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 ΩBasic 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.*

**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.