Foundations of Potential Theory
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 twofold purpose: first, to serve as an introduction for students whose attainment...
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Format:  eBook 
Language:  English 
Published: 
Berlin, Heidelberg
Springer Berlin Heidelberg
1967, 1967

Edition:  1st ed. 1967 
Series:  Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics

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Collection:  Springer Book Archives 2004  Collection details see MPG.ReNa 
Table of Contents:
 I. The Force of Gravity.
 1. The Subject Matter of Potential Theory
 2. Newton’s Law
 3. Interpretation of Newton’s Law for Continuously Distributed Bodies.
 4. Forces Due to Special Bodies
 5. Material Curves, or Wires
 6. Material Surfaces or Laminas
 7. Curved Laminas
 8. Ordinary Bodies, or Volume Distributions
 9. The Force at Points of the Attracting Masses
 10. Legitimacy of the Amplified Statement of Newton’s Law; Attraction between Bodies
 11. Presence of the Couple; Centrobaric Bodies; Specific Force
 II. Fields of Force.
 1. Fields of Force and Other Vector Fields
 2. Lines of Force
 3. Velocity Fields
 4. Expansion, or Divergence of a Field
 5. The Divergence Theorem
 6. Flux of Force; Solenoidal Fields
 7. Gauss’ Integral
 8. Sources and Sinks
 9. General Flows of Fluids; Equation of Continuity
 III. The Potential.
 1. Work and Potential Energy
 2. Equipotential Surfaces
 6. The Potential of the Solid Homogeneous Ellipsoid
 7. Remarks on the Analytic Continuation of Potentials
 8. Further Examples Leading to Solutions of Laplace’s Equation
 9. Electrostatics; Nonhomogeneous Media
 VIII.Harmonic Functions
 1. Theorems of Uniqueness
 2. Relations on the Boundary between Pairs of Harmonic Functions
 3. Infinite Regions
 4. Any Harmonic Function is a Newtonian Potential
 5. Uniqueness of Distributions Producing a Potential
 6. Further Consequences of Green’s Third Identity
 7. The Converse of Gauss’ Theorem
 IX. Electric Images; Green’s Function.
 1. Electric Images
 2. Inversion; Kelvin Transformations
 3. Green’s Function
 4. Poisson’s Integral; Existence Theorem for the Sphere
 5. Other Existence Theorems
 X. Sequences of Harmonic Functions.
 1. Harnack’s First Theorem on Convergence
 2. Expansions in Spherical Harmonics
 3. Series of Zonal Harmonics
 1. Derivatives; Laplace’s Equation
 2. Development of Potentials in Series
 3. Legendre Polynomials
 4. Analytic Character of Newtonian Potentials
 5. Spherical Harmonics
 6. Development in Series of Spherical Harmonics
 7. Development Valid at Great Distances
 8. Behavior of Newtonian Potentials at Great Distances
 VI. Properties of Newtonian Potentials at Points Occupied by Masses.
 1. Character of the Problem
 2. Lemmas on Improper Integrals
 3. The Potentials of Volume Distributions
 4. Lemmas on Surfaces
 5. The Potentials of Surface Distributions
 6. The Potentials of Double Distributions
 7. The Discontinuities of Logarithmic Potentials
 VII. Potentials as Solutions of Laplace’s Equation; Electrostatics.
 1. Electrostatics in Homogeneous Media
 2. The Electrostatic Problem for a Spherical Conductor
 3. General Coördinates
 4. Ellipsoidal Coördinates
 5. The Conductor Problem for the Ellipsoid
 13. Further Consideration of the Dirichlet Problem; Superharmonic and Subharmonic Functions
 14. Approximation to a Given Domain by the Domains of a Nested Sequence
 15. The Construction of a Sequence Defining the Solution of the Dirichlet Problem
 16. Extensions; Further Properties of U
 17. Barriers
 18. The Construction of Barriers
 19. Capacity
 20. Exceptional Points
 XII. The Logarithmic Potential.
 1. The Relation of Logarithmic to Newtonian Potentials.
 2. Analytic Functions of a Complex Variable
 3. The CauchyRiemann Differential Equations
 4. Geometric Significance of the Existence of the Derivative
 5. Cauchy’s Integral Theorem
 6. Cauchy’s Integral
 7. The Continuation of Analytic Functions
 8. Developments in Fourier Series
 9. The Convergence of Fourier Series
 10. Conformal Mapping
 11. Green’s Function for Regions of the Plane
 12. Green’s Function and Conformal Mapping
 13. The Mapping of Polygons
 4. Convergence on the Surface of the Sphere
 5. The Continuation of Harmonic Functions
 6. Harnack’s Inequality and Second Convergence Theorem
 7. Further Convergence Theorems
 8. Isolated Singularities of Harmonic Functions
 9. Equipotential Surfaces
 XI. Fundamental Existence Theorems.
 1. Historical Introduction
 2. Formulation of the Dirichlet and Neumann Problems in Terms of Integral Equations
 3. Solution of Integral Equations for Small Values of the Parameter
 4. The Resolvent
 5. The Quotient Form for the Resolvent
 6. Linear Dependence; Orthogonal and Biorthogonal Sets of Functions .
 7. The Homogeneous Integral Equations
 8. The Nonhomogeneous Integral Equation; Summary of Results for Continuous Kernels
 9. Preliminary Study of the Kernel of Potential Theory
 10. The Integral Equation with Discontinuous Kernel
 11. The Characteristic Numbers of the Special Kernel
 12. Solution of the Boundary Value Problems
 3. Potentials of Special Distributions
 4. The Potential of a Homogeneous Circumference
 5. Two Dimensional Problems; The Logarithmic Potential
 6. Magnetic Particles
 7. Magnetic Shells, or Double Distributions
 8. Irrotational Flow
 9. Stokes’ Theorem
 10. Flow of Heat
 11. The Energy of Distributions
 12. Reciprocity; Gauss’ Theorem of the Arithmetic Mean
 IV. The Divergence Theorem.
 1. Purpose of the Chapter
 2. The Divergence Theorem for Normal Regions
 3. First Extension Principle
 4. Stokes’ Theorem
 5. Sets of Points
 6. The HeineBorel Theorem
 7. Functions of One Variable; Regular Curves
 8. Functions of Two Variables; Regular Surfaces
 9. Functions of Three Variables
 10. Second Extension Principle; The Divergence Theorem for Regular Regions
 11. Lightening of the Requirements with Respect to the Field
 12. Stokes’ Theorem for Regular Surfaces
 V. Properties of Newtonian Potentials at Points of Free Space.