Difference between revisions of "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|potential theory]]: the [[Simple-layer potential|simple-layer potential]] | 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|potential theory]]: the [[Simple-layer potential|simple-layer potential]] | ||
| − | $$V(x)=\int\limits_S\frac{\mu(y)}{|y-x|}dS_y$$ | + | $$V(x)=\int\limits_S\frac{\mu(y)}{|y-x|}\,dS_y$$ |
produced by a measure distributed on a surface $S$ with density $\mu(y)$, $y\in S$; and the [[Double-layer potential|double-layer potential]] | produced by a measure distributed on a surface $S$ with density $\mu(y)$, $y\in S$; and the [[Double-layer potential|double-layer potential]] | ||
| − | $$W(x)=\int\limits_S\nu(y)\frac{\partial}{\partial n_y}\frac{1}{|y-x|}dS_y$$ | + | $$W(x)=\int\limits_S\nu(y)\frac{\partial}{\partial n_y}\frac{1}{|y-x|}\,dS_y$$ |
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|Multi-pole potential]]). | 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|Multi-pole potential]]). | ||
Latest revision as of 14:25, 14 February 2020
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).
Comments
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://encyclopediaofmath.org/index.php?title=Surface_potential&oldid=44654
Surface potential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Surface_potential&oldid=44654
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article