Foundations of Potential Theory

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.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.