Author | Kellogg, Oliver Dimon. author |
---|---|
Title | Foundations of Potential Theory [electronic resource] / by Oliver Dimon Kellogg |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1967 |
Connect to | http://dx.doi.org/10.1007/978-3-642-86748-4 |
Descript | X, 386 p. online resource |
I. The Force of Gravity. -- 1. The Subject Matter of Potential Theory -- 2. Newtonโs Law -- 3. Interpretation of Newtonโs 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 Newtonโs 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. Gaussโ Integral -- 8. Sources and Sinks -- 9. General Flows of Fluids; Equation of Continuity -- III. The Potential. -- 1. Work and Potential Energy -- 2. Equipotential Surfaces -- 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. Stokesโ Theorem -- 10. Flow of Heat -- 11. The Energy of Distributions -- 12. Reciprocity; Gaussโ 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. Stokesโ 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. 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. Stokesโ Theorem for Regular Surfaces -- V. Properties of Newtonian Potentials at Points of Free Space. -- 1. Derivatives; Laplaceโs 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 Laplaceโs Equation; Electrostatics. -- 1. Electrostatics in Homogeneous Media -- 2. The Electrostatic Problem for a Spherical Conductor -- 3. General Coรถrdinates -- 4. Ellipsoidal Coรถrdinates -- 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 Laplaceโs Equation -- 9. Electrostatics; Non-homogeneous 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 Greenโs Third Identity -- 7. The Converse of Gaussโ Theorem -- IX. Electric Images; Greenโs Function. -- 1. Electric Images -- 2. Inversion; Kelvin Transformations -- 3. Greenโs Function -- 4. Poissonโs Integral; Existence Theorem for the Sphere -- 5. Other Existence Theorems -- X. Sequences of Harmonic Functions. -- 1. Harnackโs 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. Harnackโs 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 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 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 Cauchy-Riemann Differential Equations -- 4. Geometric Significance of the Existence of the Derivative -- 5. Cauchyโs Integral Theorem -- 6. Cauchyโs Integral -- 7. The Continuation of Analytic Functions -- 8. Developments in Fourier Series -- 9. The Convergence of Fourier Series -- 10. Conformal Mapping -- 11. Greenโs Function for Regions of the Plane -- 12. Greenโs Function and Conformal Mapping -- 13. The Mapping of Polygons