Foundations of Potential Theory

Chapter 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 Chapter 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 Chapter III. The Potential. 1. Work and Potential Energy 2. Equipotential Surfaces 3. Potentials of Special Distributions 4. The Potential of a Homogenous 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 Chapter 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 Heine-Borel Theorem 7. Functions of One Variable; Regular Curves 8. Functions of Two Variables; Regular Surfaces 9. Function 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 Chapter V. Properties of Newtonian Potentials at Points of Free Space. 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 Distance 8. Behavior of Newtonian Potentials at Great Distances Chapter 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 Chapter 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 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 Equations 9. Electrostatics; Non-homogeneous Media Chapter 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 Distributins Producing a Potential 6. Further Consequences of Green's Third Identity 7. The Converse of Gauss' Theorem Chapter IX. Electric Images; Green's Function. 1. Electric Images 2. Inversion; Kelvin Tranformations 3. Green's Function 4. Poisson's Integral; Existence Theorem for the Sphere 5. Other Existence Theorems Chapter X. Sequences of Harmonic Functions. 1. Harnack's First Theorem on Convergence 2. Expansions in Spherical Harmonics 3. Series of Zonal Harmonics 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 Chapter 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 Non-homogeneous 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 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 Propeties of U 17. Barriers 18. The Construction of Barriers 19. Capacity 20. Exceptional Points Chapter XII. The Logarithmic Potential. 1. The Relation of Logarithmic to Newtonian Potentials 2. Analytic Functions of a Complex Variable 3. The Cauchy-Riemann 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 Function 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 Bibliographical Notes Index