Oliver Dimon Kellogg

Foundations of Potential Theory (eBook, PDF)

Redaktion: Courant, R.

• Format: PDF

Produktdetails

• Verlag: Springer Berlin Heidelberg
• Seitenzahl: 384
• Erscheinungstermin: 11. November 2013
• Englisch
• ISBN-13: 9783642908507
• Artikelnr.: 53156747

Inhaltsangabe

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.- 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 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. Stokes'Theorem for Regular Surfaces.- 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 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.- 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; 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 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. OtherExistence Theorems.- 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.- 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. 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.