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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]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091400/s0914001.png" /></td> </tr></table>
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$$V(x)=\int\limits_S\frac{\mu(y)}{|y-x|}dS_y$$
  
produced by a measure distributed on a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091400/s0914002.png" /> with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091400/s0914003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091400/s0914004.png" />; and the [[Double-layer potential|double-layer potential]]
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produced by a measure distributed on a surface $S$ with density $\mu(y)$, $y\in S$; and the [[Double-layer potential|double-layer potential]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091400/s0914005.png" /></td> </tr></table>
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$$W(x)=\int\limits_S\nu(y)\frac{\partial}{\partial n_y}\frac{1}{|y-x|}dS_y$$
  
produced by a measure distributed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091400/s0914006.png" /> with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091400/s0914007.png" />. Physically, a simple-layer potential is interpreted as the potential of electrical charges with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091400/s0914008.png" />, and a double-layer potential is the potential of dipoles with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091400/s0914009.png" /> (see also [[Multi-pole potential|Multi-pole potential]]).
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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]]).
  
  

Revision as of 14:25, 23 August 2014

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=12154
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article