# Double layer potential

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Double_layer_potential
> Markdown URL: https://mediated.wiki/source/Double_layer_potential.md
> Source: https://en.wikipedia.org/wiki/Double_layer_potential
> Source revision: 1084982907
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [potential theory](/source/potential_theory), an area of [mathematics](/source/mathematics), a '''double layer potential''' is a solution of [Laplace's equation](/source/Laplace's_equation) corresponding to the [electrostatic](/source/electrostatic_potential) or [magnetic potential](/source/magnetic_scalar_potential) associated to a [dipole](/source/dipole) distribution on a closed surface ''S'' in three-dimensions.  Thus a double layer potential {{math|''u''('''x''')}} is a scalar-valued function of {{math|'''x''' ∈ '''R'''<sup>3</sup>}} given by
<math display="block">u(\mathbf{x}) = \frac {-1} {4\pi} \int_S \rho(\mathbf{y}) \frac{\partial}{\partial\nu} \frac{1}{|\mathbf{x}-\mathbf{y}|} \, d\sigma(\mathbf{y})</math>
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](/source/hypersurface) ''S'' in ''n''-dimensional [Euclidean space](/source/Euclidean_space) by means of
<math display="block">u(\mathbf{x}) = \int_S \rho(\mathbf{y})\frac{\partial}{\partial\nu} P(\mathbf{x}-\mathbf{y})\,d\sigma(\mathbf{y})</math>
where ''P''('''y''') is the [Newtonian kernel](/source/Newtonian_kernel) in ''n'' dimensions.

==See also==
*[Single layer potential](/source/Single_layer_potential)
*[Potential theory](/source/Potential_theory)
*[Electrostatics](/source/Electrostatics)
*[Laplacian of the indicator](/source/Laplacian_of_the_indicator)

==References==
* {{citation|first1=Richard|last1=Courant|authorlink1=Richard Courant|first2=David|last2=Hilbert|authorlink2=David Hilbert|title=Methods of Mathematical Physics, Volume II|publisher=Wiley-Interscience|year=1962}}.
* {{Citation | last1=Kellogg | first1=O. D. | title=Foundations of potential theory | publisher=[Dover Publications](/source/Dover_Publications) | location=New York | isbn=978-0-486-60144-1 | year=1953}}.
* {{springer|id=d/d033880|title=Double-layer potential|first=I.A.|last=Shishmarev}}.
* {{springer|id=m/m065210|title=Multi-pole potential|first=E.D.|last=Solomentsev}}.

Category:Potential theory

---
Adapted from the Wikipedia article [Double layer potential](https://en.wikipedia.org/wiki/Double_layer_potential) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Double_layer_potential?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.