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 Sur |
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. Bar |
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 |