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 {{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 ''S'' in ''n''-dimensional 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 in ''n'' dimensions.
==See also== *Single layer potential *Potential theory *Electrostatics *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 | 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