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Surface potential

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The potential of a measure that is concentrated on a certain surface. Two forms of a surface potential are used in handling major boundary value problems in potential theory: the simple-layer potential

$$V(x)=\int\limits_S\frac{\mu(y)}{|y-x|}dS_y$$

produced by a measure distributed on a surface $S$ with density $\mu(y)$, $y\in S$; and the double-layer potential

$$W(x)=\int\limits_S\nu(y)\frac{\partial}{\partial n_y}\frac{1}{|y-x|}dS_y$$

produced by a measure distributed on $S$ with density $\nu(y)$. Physically, a simple-layer potential is interpreted as the potential of electrical charges with density $\mu(y)$, and a double-layer potential is the potential of dipoles with density $\nu(y)$ (see also Multi-pole potential).


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References

[a1] O.D. Kellogg, "Foundations of potential theory" , Dover, reprint (1953) (Re-issue: Springer, 1967)
How to Cite This Entry:
Surface potential. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Surface_potential&oldid=33112
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article