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 two fundamental forces of nature known at the time, namely gra ...
, an area of
mathematics, a double layer potential is a solution of
Laplace's equation corresponding to the
electrostatic
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amb ...
or
magnetic potential associated to a
dipole
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
*An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system ...
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
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
''S'' in ''n''-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
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 two fundamental forces of nature known at the time, namely gra ...
*
Electrostatics
*
Laplacian of the indicator
In mathematics, the Laplacian of the indicator of the domain ''D'' is a generalisation of the derivative of the Dirac delta function to higher dimensions, and is non-zero only on the ''surface'' of ''D''. It can be viewed as the ''surface delta pr ...
References
* .
* .
* .
* {{springer, id=m/m065210, title=Multi-pole potential, first=E.D., last=Solomentsev.
Potential theory