In
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely g ...
, an area of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a double layer potential is a solution of
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delt ...
corresponding to the
electrostatic
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word (), mean ...
or
magnetic potential associated to a
dipole
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
* An electric dipole moment, electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple ...
distribution on a closed surface ''S'' in three-dimensions. Thus a double layer potential is a scalar-valued function of 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
by means of
where ''P''(y) is the
Newtonian kernel in ''n'' dimensions.
See also
*
Single layer potential
*
Potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely g ...
*
Electrostatics
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.
Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...
*
Laplacian of the indicator
References
* .
* .
* .
* {{springer, id=m/m065210, title=Multi-pole potential, first=E.D., last=Solomentsev.
Potential theory