Foundations of Potential Theory

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Springer Science & Business Media, 11 nov. 2013 - 384 pages
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The present volume gives a systematic treatment of potential functions. It takes its origin in two courses, one elementary and one advanced, which the author has given at intervals during the last ten years, and has a two-fold purpose: first, to serve as an introduction for students whose attainments in the Calculus include some knowledge of partial derivatives and multiple and line integrals; and secondly, to provide the reader with the fundamentals of the subject, so that he may proceed immediately to the applications, or to the periodical literature of the day. It is inherent in the nature of the subject that physical intuition and illustration be appealed to freely, and this has been done. However, in order that the book may present sound ideals to the student, and also serve the mathematician, both for purposes of reference and as a basis for further developments, the proofs have been given by rigorous methods. This has led, at a number of points, to results either not found elsewhere, or not readily accessible. Thus, Chapter IV contains a proof for the general regular region of the divergence theorem (Gauss', or Green's theorem) on the reduction of volume to surface integrals. The treatment of the fundamental existence theorems in Chapter XI by means of integral equations meets squarely the difficulties incident to the discontinuity of the kernel, and the same chapter gives an account of the most recent developments with respect to the Dirichlet problem.
 

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Table des matières

Material Curves or Wires
8
Ordinary Bodies or Volume Distributions
15
Legitimacy of the Amplified Statement of Newtons Law Attraction
22
Chapter II
28
Chapter IV
39
Sources and Sinks
44
Potentials of Special Distributions
55
Magnetic Particles
65
The Converse of Gauss Theorem
224
Chapter IX
228
Inversion Kelvin Transformations
231
Greens Function
236
Poissons Integral Existence Theorem for the Sphere
240
Other Existence Theorems
244
Chapter X
248
Expansions in Spherical Harmonics
251

Stokes Theorem
72
The Energy of Distributions
79
Purpose of the Chapter
84
The Divergence Theorem for Normal Regions
85
First Extension Principle
88
Stokes Theorem
89
Sets of Points
91
The HeineBorel Theorem
94
Functions of One Variable Regular Curves
97
Functions of Two Variables Regular Surfaces
100
Functions of Three Variables
113
Lightening of the Requirements with Respect to the Field
119
Stokes Theorem for Regular Surfaces
121
Development of Potentials in Series
124
Legendre Polynomials
125
Analytic Character of Newtonian Potentials
135
Spherical Harmonics
139
Development in Series of Spherical Harmonics
141
Development Valid at Great Distances
143
Behavior of Newtonian Potentials at Great Distances
144
The Logarithmic Potential
145
Chapter VI
146
The Potentials of Volume Distributions
150
Lemmas on Surfaces
157
The Potentials of Surface Distributions
160
The Potentials of Double Distributions
166
The Discontinuities of Logarithmic Potentials
172
Chapter VII
175
The Electrostatic Problem for a Spherical Conductor
176
General Coördinates
178
Ellipsoidal Coördinates
184
The Conductor Problem for the Ellipsoid
188
The Potential of the Solid Homogeneous Ellipsoid
192
Remarks on the Analytic Continuation of Potentials
196
Further Examples Leading to Solutions of Laplaces Equation
198
Electrostatics Nonhomogeneous Media
206
Chapter VIII
211
Relations on the Boundary between Pairs of Harmonic Functions
215
Infinite Regions
216
Any Harmonic Function is a Newtonian Potential
218
Uniqueness of Distributions Producing a Potential
220
Further Consequences of Greens Third Identity
223
Series of Zonal Harmonics
254
Convergence on the Surface of the Sphere
256
The Continuation of Harmonic Functions
259
Harnacks Inequality and Second Convergence Theorem
262
Further Convergence Theorems
264
Isolated Singularities of Harmonic Functions
268
Equipotential Surfaces
273
Chapter XI
277
Formulation of the Dirichlet and Neumann Problems in Terms of Inte gral Equations
286
Solution of Integral Equations for Small Values of the Parameter
287
The Resolvent
289
The Quotient Form for the Resolvent
290
Linear Dependence Orthogonal and Biorthogonal Sets of Functions
292
The Homogeneous Integral Equations
294
The Nonhomogeneous Integral Equation Summary of Results for Con tinuous Kernels
297
Preliminary Study of the Kernel of Potential Theory
299
The Integral Equation with Discontinuous Kernel
307
The Characteristic Numbers of the Special Kernel
309
Solution of the Boundary Value Problems
311
Further Consideration of the Dirichlet Problem Superharmonic and Subharmonic Functions
315
Approximation to a Given Domain by the Domains of a Nested Sequence
317
The Construction of a Sequence Defining the Solution of the Dirichlet Problem
322
Extensions Further Properties of U
323
Barriers
326
The Construction of Barriers
328
Capacity
330
Exceptional Points 834
334
The Relation of Logarithmic to Newtonian Potentials
338
Analytic Functions of a Complex Variable
340
The CauchyRiemann Differential Equations
341
Geometric Significance of the Existence of the Derivative
343
Cauchys Integral Theorem
344
Cauchys Integral
348
The Continuation of Analytic Functions
351
Developments in Fourier Series
353
The Convergence of Fourier Series
355
Conformal Mapping
359
Greens Function for Regions of the Plane
363
Greens Function and Conformal Mapping
365
The Mapping of Polygons
370
Bibliographical Notes
377
Index
379

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