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Foundations of Potential Theory

Oliver Dimon Kellogg
Dover Publications
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[Reviewed by
William J. Satzer
, on

This is an unaltered republication by Dover Books of a book that was first published in 1929. The term “potential theory” comes from nineteenth century physics and the belief that the fundamental forces in nature were derived from potential functions that satisfy Laplace’s equation. While this is true for classical electrostatics and Newtonian gravitation, physics has moved on to more comprehensive nonlinear theories for which Laplace’s equation no longer suffices. The classical potential theory that developed from these origins in physics focused on the properties of harmonic functions. Since the first publication of this book the study of potential theory has continued in several directions, including significant applications in probability (Doob’s work, for example), as well as investigations of the consequences of the conformal symmetries of Laplace’s equation.

Kellogg’s book has two components. The first focuses on mathematical physics, and specifically on applications in gravitation and electrostatics. This part has more immediate connections with physical intuition; it is more elementary and accessible to readers with backgrounds that include some experience with partial derivatives, line and multiple integrals. The second component looks more deeply into fundamentals, uses more rigorous methods, and includes a significant number of proofs. The two parts of the book are of roughly equal length, and blend smoothly from one into the other. Green’s function, for example, is introduced as a tool for solving the Dirichlet problem — determining whether a harmonic function exists on a closed region that takes pre-assigned continuous boundary values. But this is done first in the context of electric charge on a grounded surface.

Kellogg’s was one of the first textbooks in potential theory, and he does well by the subject. He provides a path for an advanced undergraduate or beginning graduate student to go from fairly concrete and intuitive physical examples to a rigorous treatment of the theory of harmonic functions. Students now would probably look to more modern treatments for the formal development, but would benefit from the careful, intuitively-based introduction to the subject.

Kellogg has a good sense of the reader, a pleasant style, and he’s surprising entertaining to read.


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