Table Of ContentPURE AND APPLIED MATHEMATICS
A SERIES OF MONOGRAPHS AND TEXTBOOKS
Geomathematically Oriented
Potential Theory
Willi Freeden
Christian Gerhards
Geomathematically Oriented
Potential Theory
PURE AND APPLIED MATHEMATICS
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Geomathematically Oriented
Potential Theory
Willi Freeden
University of Kaiserslautern
Germany
Christian Gerhards
University of Kaiserslautern
Germany
CRC Press
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Contents
Preface ix
About the Authors xiii
List of Symbols xv
Introduction 1
I Preliminaries 9
1 Three-Dimensional Euclidean Space R3 11
1.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Two–Dimensional Sphere Ω 29
2.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 (Scalar) Spherical Harmonics . . . . . . . . . . . . . . . . . . 42
2.4 (Scalar) Circular Harmonics . . . . . . . . . . . . . . . . . . 52
2.5 Vector Spherical Harmonics . . . . . . . . . . . . . . . . . . . 59
2.6 Tensor Spherical Harmonics . . . . . . . . . . . . . . . . . . 69
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
II Potential Theory in the Euclidean Space R3 81
3 Basic Concepts 83
3.1 Background Material . . . . . . . . . . . . . . . . . . . . . . 83
3.2 Volume Potentials . . . . . . . . . . . . . . . . . . . . . . . . 99
3.3 Surface Potentials . . . . . . . . . . . . . . . . . . . . . . . . 103
3.4 Boundary-Value Problems . . . . . . . . . . . . . . . . . . . 126
3.5 Locally and Globally Uniform Approximation . . . . . . . . 152
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4 Gravitation 175
4.1 Oblique Derivative Problem . . . . . . . . . . . . . . . . . . 181
4.2 Satellite Problems . . . . . . . . . . . . . . . . . . . . . . . . 212
vii
viii Contents
4.3 Gravimetry Problem . . . . . . . . . . . . . . . . . . . . . . . 224
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
5 Geomagnetism 243
5.1 Geomagnetic Background . . . . . . . . . . . . . . . . . . . . 243
5.2 Mie and Helmholtz Decompositions . . . . . . . . . . . . . . 248
5.3 Gauss Representation and Uniqueness . . . . . . . . . . . . . 256
5.4 Separation of Sources . . . . . . . . . . . . . . . . . . . . . . 266
5.5 Ionospheric Current Systems . . . . . . . . . . . . . . . . . . 272
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
III Potential Theory on the Unit Sphere Ω 285
6 Basic Concepts 287
6.1 Background Material . . . . . . . . . . . . . . . . . . . . . . 287
6.2 Surface Potentials . . . . . . . . . . . . . . . . . . . . . . . . 293
6.3 Curve Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 297
6.4 Boundary-Value Problems . . . . . . . . . . . . . . . . . . . 316
6.5 Differential Equations for ∇∗ and L∗ . . . . . . . . . . . . . 333
6.6 Locally and Globally Uniform Approximation . . . . . . . . 336
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
7 Gravitation 347
7.1 Disturbing Potential . . . . . . . . . . . . . . . . . . . . . . . 347
7.2 Linear Regularization Method . . . . . . . . . . . . . . . . . 359
7.3 Multiscale Solution . . . . . . . . . . . . . . . . . . . . . . . 363
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
8 Geomagnetism 385
8.1 Mie and Helmholtz Decomposition . . . . . . . . . . . . . . . 385
8.2 Higher-Order Regularization Methods . . . . . . . . . . . . . 395
8.3 Separation of Sources . . . . . . . . . . . . . . . . . . . . . . 404
8.4 Ionospheric Current Systems . . . . . . . . . . . . . . . . . . 411
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
Bibliography 429
Index 449
Preface
In the year 1782, P.S. Laplace showed that the Newtonian potential obeys a
certainpartialdifferentialequation.Laplace’sequation–asitiscalledtoday–
hasbecome a centraldifferentialequationinarguablyallphysicalgeosciences
because of the wide rangeof phenomena that it characterizes.It is important
in gravitation,electromagnetism, and fluid dynamics because it describes the
behaviorofgravitational,magnetic,andfluidpotentials.Thetheoryofpoten-
tials concerned with the Laplace equation is known as potential theory. It is
thesubjectofthistextbook,butwithparticularemphasisontheEarth’sgrav-
itational and magnetic field. Our understanding of potential theory, however,
doesnotintendtoexplainthe natureofgravitationormagnetism.Thisisthe
taskofphysics;itrequiresadetailedstudyofNewton’sandEinstein’stheories
of gravitation and Maxwell’s theory to magnetism as well as modern devel-
opments in these fields. Our work rather intends to gain some insight on how
gravitationandmagnetismcanbegeomathematicallyhandledfortherelevant
observables and on how the resulting geopotential problems can be solved in
asystematic,mathematicallyrigorousframeworkinvolvingthree-dimensional
Euclidean as well as spherical concepts.
Consideringthehugeliteratureingeosciencesonthegravitationalandthe
magnetic field, it might be asked why a new textbook on potential theory is
neededatthistime.Indeed,classicalpotentialtheoryisawell-understoodand
frequentlydocumentedareainmathematics.Itdealswithharmonicfunctions,
maximum/minimum principles, single and multipole expansions,Green func-
tions,integralrepresentationsofpotentials,boundary-valueproblems,volume
and surface integral equation methods, etc. The essential texts on potential
theorypublishedduringthelastcenturyarestillavailabletoday,notablythose
of O.D. Kellogg [1967], L.L. Helms [1969], and R.J. Blakely [1996]. These
books deal thoroughly with the fundamentals of potential theory in three-
dimensional Euclidean space, more precisely in the interior and/or exterior
of “potato-like” bodies such as a ball, ellipsoid, geoid, and the actual Earth.
Even more, multiscale methods for boundary-value and inverse problems (cf.
W. Freeden, V. Michel [2004]) have been studied in more detail.
Nevertheless, we believe that this book on geomathematically oriented
potential theory will fill a significant gap. As mathematicians interested in
any kind of Earth-related gravitational and magnetic data and processes, we
immediately found ourselves involved with the Laplace operator in specific
application to functions defined on surfaces in three-dimensional Euclidean
space.It seemedto us, andwe also find ittrue today,that no single textbook
ix
