Foundations of potential theory

Oliver D. Kellogg

Introduction to fundamentals of potential functions covers: the force of gravity, fields of force, potentials, harmonic functions, electric images and Green’s function, sequences of harmonic functions, fundamental existence theorems, the logarithmic potential and much more. Detailed proofs rigorously worked out. 1929 ed.

---

Table of contents :
Title page ……Page 1
Preface ……Page 3
Contents ……Page 4
1. The Subject Matter of Potential Theory ……Page 9
2. Newton’s Law ……Page 10
3. Interpretation of Newton’s Law for Continuously Distributed Bodies ……Page 11
4. Forces Due to Special Bodies ……Page 12
5. Material Curves, or Wires ……Page 16
6. Material Surfaces or Laminas ……Page 18
7. Curved Laminas ……Page 20
8. Ordinary Bodies, or Volume Distributions ……Page 23
9. The Force at Points of the Attracting Masses ……Page 25
10. Legitimacy of the Amplified Statement of Newton’s Law; Attraction between Bodies ……Page 30
11. Presence of the Couple; Centrobaric Bodies; Specific Force ……Page 34
2. Lines of Force ……Page 36
3. Velocity Fields ……Page 39
4. Expansion, or Divergence of a Field ……Page 42
5. The Divergence Theorem ……Page 45
6. Flux of Force; Solenoidal Fields ……Page 48
7. Gauss’ Integral ……Page 50
8. Sources and Sinks ……Page 52
9. General Flows of Fluids; Equation of Continuity ……Page 53
1. Work and Potential Energy ……Page 56
2. Equipotential Surfaces ……Page 62
3. Potentials of Special Distributions ……Page 63
4. The Potential of a Homogeneous Circumference ……Page 66
5. Two Dimensional Problems; The Logarithmic Potential ……Page 70
6. Magnetic Particles ……Page 73
7. Magnetic Shells, or Double Distributions ……Page 74
8. Irrotational Flow ……Page 77
9. Stokes’ Theorem ……Page 80
10. Flow of Heat ……Page 84
11. The Energy of Distributions ……Page 87
12. Reciprocity; Gauss’ Theorem of the Arithmetic Mean ……Page 90
1. Purpose of the Chapter ……Page 92
2. The Divergence Theorem for Normal Regions ……Page 93
3. First Extension Principle ……Page 96
4. Stokes’ Theorem ……Page 97
5. Sets of Points ……Page 99
6. The Heine-Borel Theorem ……Page 102
7. Functions of One Variable; Regular Curves ……Page 105
8. Functions of Two Variables; Regular Surfaces ……Page 108
10. Second Extension Principle; The Divergence Theorem for Regular Regions ……Page 121
11. Lightening of the Requirements with Respect to the Field ……Page 127
1. Derivatives; Laplace’s Equation ……Page 129
2. Development of Potentials in Series ……Page 132
3. Legendre Polynomials ……Page 133
4. Analytic Character of Newtonian Potentials ……Page 143
5. Spherical Harmonics ……Page 147
6. Development in Series of Spherical Harmonics ……Page 149
7. Development Valid at Great Distances ……Page 151
8. Behavior ot Newtonian Potentials at Cireat Distances ……Page 152
2. Lemmas on Improper Integrals ……Page 154
3. The Potentials of Volume Distributions ……Page 158
4. Lemmas on Surfaces ……Page 165
5. The Potentials of Surface Distributions ……Page 168
6. The Potentials of Double Distributions ……Page 174
7. The Discontinuities of Logarithmic Potentials ……Page 180
1. Electrostatics in Homogeneous Media ……Page 183
2. The Electrostatic Problem for a Spherical Conductor ……Page 184
3. General Coordinates ……Page 186
4. Ellipsoidal Coordinates ……Page 192
5. The Conductor Problem for the Ellipsoid ……Page 196
6. The Potential of the Solid Homogeneous Ellipsoid ……Page 200
7. Remarks on the Analytic Continuation of Potentials ……Page 204
8. Further Examples Leading to Solutions of Laplace’s Equation ……Page 206
9. Electrostatics; Non-homogeneous Media ……Page 214
1. Theorems of Uniqueness ……Page 219
2. Relations on the Boundary between Pairs of Harmonic Functions ……Page 223
3. Infinite Regions ……Page 224
4. Any Harmonic Function is a Newtonian Potential ……Page 226
5. Uniqueness of Distributions Producing a Potential ……Page 228
6 Further Consequences of Green’s Third Identity ……Page 231
7. The Converse of Gauss’ Theorem ……Page 232
1. Electric Images ……Page 236
2. Inversion; Kelvin Transformations ……Page 239
3. Green’s Function ……Page 244
4. Poisson’s Integral; Existence Theorem for the Sphere ……Page 248
5. Other Existence Theorems ……Page 252
1. Harnack’s First Theorem on Convergence ……Page 256
2. Expansions in Spherical Harmonics ……Page 259
3. Scries of Zonal Harmonics ……Page 262
4. Convergence on the Surface of the Sphere ……Page 264
5. The Continuation of Harmonic Functions ……Page 267
6. Harnack’s Inequality and Second Convergence Theorem ……Page 270
7. Further Convergence Theorems ……Page 272
8. Isolated Singularities of Harmonic Functions ……Page 276
9. Equipotential Surfaces ……Page 281
1. Historical Introduction ……Page 285
2. Formulation of the Dirichlet and Neumann Problems in Terms of Integral Equations ……Page 294
3. Solution of Integral Equations for Small Values ol the Parameter ……Page 295
4. The Resolvent ……Page 297
5. The Quotient Form for the Resolvent ……Page 298
6. Linear Dependence; Orthogonal and Biorthogonal Sets of Functions ……Page 300
7. The Homogeneous Integral Equations ……Page 302
8. The Non-homogeneous Integral Equation; Summary of Results for Continuous Kernels ……Page 305
9. Preliminary Study of the Kernel of Potential Theory ……Page 307
10. The Integral Equation with Discontinuous Kernel ……Page 315
11. The Characteristic Numbers of the Special Kernel ……Page 317
12. Solution of the Boundary Value Problems ……Page 319
13. Further Consideration of the Dirichlet Problem; Superharmonic and Subharmonic Functions ……Page 323
14. Approximation to a Given Domain by the Domains of a Nested Sequence ……Page 325
15. The Construction of a Sequence Defining the Solution of the Dirichlet Problem ……Page 330
16. Extensions; Further Properties of $U$ ……Page 331
17. Barriers ……Page 334
18. The Construction of Barriers ……Page 336
19. Capacity ……Page 338
20. Exceptional Points ……Page 342
1. The Relation of Logarithmic to Newtonian Potentials ……Page 346
2. Analytic Functions of a Complex Variable ……Page 348
3. The Cauchy-Riemann Differential Equations ……Page 349
4. Geometric Significance of the Existence of the Derivative ……Page 351
5. Cauchy’s Integral Theorem ……Page 352
6. Cauchy’s Integral ……Page 356
7. The Continuation of Analytic Functions ……Page 359
8. Developments in Fourier Scries ……Page 361
9. The Convergence of Fourier Series ……Page 363
10. Conformal Mapping ……Page 367
11. Green’s Function for Regions of the Plane ……Page 371
12. Green’s Function and Conformal Mapping ……Page 373
13. The Mapping of Polygons ……Page 378
Bibliographical Notes ……Page 385
Index ……Page 387