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Foundations of potential theory
Oliver Dimon Kellogg;

Titelseite

Einleitung

Tabelle, Liste

Chapter I. The Force of Gravity.
1
1. The Subject Matter of Potential Theory.
1
2. Newton's Law.
2
3. Interpretation of Newton's Law for Continously Distributed Bodies.
3
4. Forces Due to special bodies.
4
5. Material Curves or Wires.
8
6. Material Surfaces or Lamina.
10
7. Curved Laminas.
12
8. Ordinary Bodies, or Volume Distributions.
15
9. The Force at Pooints of the Attracting Masses.
17
10. Legitimacy of the amplified statement of Newton's law; attraction between bodies.
22
11. Presence of the couple; centorbaric bodies; specific force.
26
Chapter II. Fields of Force.
28
1. Fields of Force and other vector fields.
28
2. Lines of Force.
28
3. Velocity Fields.
31
4. Expansion, or divergence of a field.
34
5. The divergence theorem.
37
6. Flux of Force; Solenoidal fields.
40
7. Gauss' integral.
42
8. Sources and sinks.
44
9. General flows of fluids; equation of continuity.
45
Chapter III. The Potential.
48
1. Work and Potential energy.
48
2. Equipotential surfaces.
54
3. Potentials of Special distributions.
55
4. The Potential of a Homogeneous circumference.
58
5. Two Dimensional Problems.
62
6. Magnetic Particles.
65
7. Magnetic Shells, or Double Distributions.
66
8. Irrotaional Flow.
69
9. Stoke's Theorem.
72
10. Flow of Heat.
76
11. The Energy of Distributions.
79
12. Reciprocity; Gauss' Theorem of the Arithmetic Mean.
82
Chapter IV. The Divergence Theorem.
84
1. Purpose of the Chapter.
84
2. The Divergence Theorem ofr normal Regions.
85
3. First Extension principle.
88
4. Stoke's theorem.
89
5. Sets of Points.
91
6. The HeineBorel Theorem.
94
7. Functions of One Variable; Regular Curves.
97
8. Functions of Two Variables; Regular Surfaces.
100
9. Functions of three variables.
113
10. Second Extension principle; the divergence theorem for regular regions.
113
11. Lightening of the requirements with respect to the Field.
119
12. Stoke's Theorem for Regular Surfaces.
121
Chapter V. Properties of Newtonian Potentials at points of free space.
121
1. Derivatives; Laplace'es Equation.
121
2. Development of Potentials in Series.
124
3. Legendre Polynomials.
125
4. Analytic Character of Newtonian Potentials.
135
5. Spherical Harmonics.
139
6. Development in Series of Spherical Harmonics.
141
7. Development Valid at Great Distances.
143
8. Behavior of Newtonian Potentials at Great Distances.
144
Chapter VI. Properties of Newtonian Potentials at Points Occupied by Masses.
146
1. Character of the Problem.
146
2. Lemmas on Improper Integrals.
146
3. The Potentials of Volume Distribution.
150
4. Lemmas on Surfaces.
157
5. The Potentials of Surface Distribution.
160
6. The Potentials of Double Distributions.
166
7. The Discontinuities of Logarithmic Potentials.
172
Chapter VII. Potentials as Solutions of Laplace's Equation; Electrostatics.
175
1. Electrostatics in Homogeneous Media.
175
2. The Electrostatic Problem for a Spherical Conductor.
176
3. General Coordinates.
178
4. Ellipsoidal Coordinates.
184
5. The Conductor Problem for the Ellipsoid.
188
6. The Potential of the solid homogeneous ellipsoid.
192
7. Remarks on the analytic continuation of potentials.
196
8. Further Examples leading to solutions of Laplace's equation.
198
9. Electrostatics; nonhomogeneous media.
206
Chapter VIII. Harmonic Functions.
211
1. Theorems of Uniqueness.
211
2. Relations on the Boundary between pairs of Harmonic functions.
215
3. Infinite Regions.
216
4. Any harmonic function is a Newtonian Potential.
218
5. Uniqueness of Distributions producing a potential.
220
6. Further consequences of Green's Third identity.
223
7. The Converse of Gauss' Theorem.
224
Chapter IX. Electric Image; Green's Function.
228
1. Electric Images.
228
2. Inversion; Kelvin Transformation.
231
3. Green's Function.
236
4. Poisson's Integral; Existence theorem for the sphere.
240
5. Other existence theorems.
244
Chapter X. Sequences of harmonic functions.
248
1. Harnack's First Theorem on Convergence.
248
2. Expansions in Spherical Harmonics.
251
3. Series of Zonal Harmonics.
254
4. Convergence on the surface of the sphere.
256
5. The continuation of harmonic functions.
259
6. Harnack's Inequality and second convergence theorem.
262
7. Further convergence theorems.
264
8. Isolated Singularities of Harmonic Functions.
268
9. Equipotential surfaces.
273
Chapter XI. Fundamental Existence Theorems.
277
1. Historical Introduction.
277
2. Formulation of the Dirchlet and Neumann Problems in Terms of Integral Equations.
286
3. Solution of Integral Equations for Small Values of the Parameter.
287
4. The Resolvent.
289
5. The Quotient Form for the Resolvent.
290
6. Linear Dpendence; Orthogonal and Biorthogonal Sets of Functions.
292
7. The Homegenous Integral Equations.
294
8. The Nonhomogeneous Integral Equation; summary of results of continous Kernels.
297
9. Preliminary study of the Kernel of Potential Theory.
299
10. The Integral Equation with Discontinuous Kernel.
307
11. The Characteristic Numbers of the special Kernel.
309
12. Solution of the Boundary Value Problems.
311
13. Further Consideration of the Dirchlet Problem; Superharmonic and Subharmonci functions.
315
14. Approximation to a Given Domain by the Domains of a Nested Sequence.
317
15. The Construction of a Sequence Definig the Solution of the Dirchlet Problem.
322
16. Extension; Futher properties of U.
323
17. Barries.
326
18. The Construction Barries.
328
19. Capacity.
330
20. Exponential Points.
334
Chapter XII. The logarithmic potential.
338
1. The relation fo logarithmic to Newtonian Potentials.
338
2. Analytic Functions of a Complex Variable.
340
3. The CauchyRiemann differential equations.
341
4. Geometric Significance of the Existence of the Derivative.
343
5. Cauchy's Integral Theorem.
344
6. Cauchy's Integral.
348
7. The Continuation of Analytic Functions.
351
8. Developments in Fourier Series.
353
9. The Convergence of Fourier Series.
355
10. Conformal Mapping.
359
11. Green's Function for Regions of the Plane.
363
12. Green's Function and Conformal Mapping.
365
13. The Mapping of Polygons.
370
Tabelle, Liste
377
Index.
379