- Double layer potential
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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.
References
- Courant, Richard; Hilbert, David (1962), Methods of Mathematical Physics, Volume II, Wiley-Interscience.
- Kellogg, O. D. (1953), Foundations of potential theory, New York: Dover Publications, ISBN 978-0-486-60144-1.
- Shishmarev, I.A. (2001), "Double-layer potential", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/d/d033880.htm.
- Solomentsev, E.D. (2001), "Multi-pole potential", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/m/m065210.htm.
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