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 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.
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