Double layer potential
Encyclopedia
In potential theory
, an area of mathematics
, a double layer potential is a solution of Laplace's equation
corresponding to the electrostatic or magnetic potential
associated to a dipole
distribution on a closed surface S in three-dimensions. Thus a double layer potential u(x) is a scalar-valued function of x ∈ R3 given by
where ρ denotes the dipole distribution, ∂/∂ν denotes the directional derivative in the direction of the outward unit normal in the y variable, and dσ is the surface measure on S.
More generally, a double layer potential is associated to a hypersurface
S in n-dimensional Euclidean space
by means of
where P(y) is the Newtonian kernel in n dimensions.
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
, an area of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a double layer potential is a solution of Laplace's equation
Laplace's equation
In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...
corresponding to the electrostatic or magnetic potential
Magnetic potential
The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, and the magnetic scalar potential, ψ...
associated to a dipole
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...
distribution on a closed surface S in three-dimensions. Thus a double layer potential u(x) is a scalar-valued function of x ∈ R3 given by
where ρ denotes the dipole distribution, ∂/∂ν denotes the directional derivative in the direction of the outward unit normal in the y variable, and dσ is the surface measure on S.
More generally, a double layer potential is associated to a hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
S in n-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
by means of
where P(y) is the Newtonian kernel in n dimensions.