In mathematics and mathematical physics, potential theory is the study of harmonic functions, the solutions of Laplace equation in both pure mathematics and its applications in physics. A central organizing idea in the field is to consider the symmetries of the Laplace equation itself, which is one useful starting point for approaching the harmonic functions that solve it.
Facts
Central QuestionPotential theory asks what properties harmonic functions share, the solutions of Laplace's and Poisson's equations that model phenomena such as the gravitational and electrostatic potential, particularly around their singularities and boundary behavior. 1 Key DebatePotential theory has no settled boundary separating it from the general theory of Poisson's equation: the two fields overlap so far that scholars distinguish them only by emphasis, counting a result about the singularities of harmonic functions as potential theory while counting a result about how a solution depends on boundary data as belonging to the theory of Poisson's equation, with no hard rule dividing the two in practice. 1 Classification
Pure or AppliedBoth / Interdisciplinary 1 Potential Theory
Filter Results5 entries
Connections
Includes
Source Gustave Choquet, Biography (MacTutor History of Mathematics)
Source Joseph L. Doob (Wikipedia)
Source Marcel Riesz, Biography (MacTutor History of Mathematics)
Source Peter Gustav Lejeune Dirichlet (Wikipedia)
Source Poppy-seed bagel theorem, Wikipedia
Source Walther von Dyck, Biography (MacTutor History of Mathematics)
Sources
1. Potential Theory (Wikipedia)
Wikimedia FoundationLead section
The term "potential theory" dates from 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which satisfy Poisson's equation-or in the vacuum, Laplace's equation.
Lead section, third paragraph
The difference is more one of emphasis than subject matter and rests on the following distinction: potential theory focuses on the properties of the functions as opposed to the properties of the equation.
Symmetry section, opening sentence
A useful starting point and organizing principle in the study of harmonic functions is a consideration of the symmetries of the Laplace equation.
Lead section, pure or applied classification
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
View the Source Peter Gustav Lejeune Dirichlet (Wikipedia)
Includes: Peter Gustav Lejeune Dirichlet, Lead paragraph [in-branch 4]Quote, Includes: Peter Gustav Lejeune Dirichlet, Lead paragraph [in-branch 4]
potential theory
Joseph L. Doob (Wikipedia)
Includes: Joseph L. Doob, Lead paragraph [in-branch 2]Quote, Includes: Joseph L. Doob, Lead paragraph [in-branch 2]
probabilistic potential theory
Poppy-seed bagel theorem, Wikipedia
Includes: Poppy-Seed Bagel Theorem, Lead paragraphQuote, Includes: Poppy-Seed Bagel Theorem, Lead paragraph
In particular, this includes the Coulomb law observed in electrostatics and Riesz potentials extensively studied in potential theory.
Gustave Choquet, Biography (MacTutor History of Mathematics)
Includes: Gustave Choquet, SummaryQuote, Includes: Gustave Choquet, Summary
Gustave Choquet was a French mathematician who worked in functional analysis, potential theory, topology and measure theory.
Marcel Riesz, Biography (MacTutor History of Mathematics)
Includes: Marcel Riesz, SummaryQuote, Includes: Marcel Riesz, Summary
Marcel Riesz was a Hungarian-born mathematician who worked on summation methods, potential theory and other parts of analysis, as well as number theory and partial differential equations.
Walther von Dyck, Biography (MacTutor History of Mathematics)
Includes: Walther von Dyck, SummaryQuote, Includes: Walther von Dyck, Summary
Von Dyck made important contributions to function theory, group theory ( where a fundamental result on group presentations is named after him ) topology and potential theory.
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