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<title>Double layer potential</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Double layer potential</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Potential_theory" title="Potential theory">potential theory</a>, an area of <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>double layer potential</b> is a solution of <a href="Laplace's_equation" title="Laplace's equation">Laplace's equation</a> corresponding to the <a href="Electrostatic_potential" class="mw-redirect" title="Electrostatic potential">electrostatic</a> or <a href="Magnetic_scalar_potential" title="Magnetic scalar potential">magnetic potential</a> associated to a <a href="Dipole" title="Dipole">dipole</a> distribution on a closed surface <i>S</i> in three-dimensions. Thus a double layer potential <span class="texhtml"><i>u</i>(<b>x</b>)</span> is a scalar-valued function of <span class="texhtml"><b>x</b> ∈ <b>R</b><sup>3</sup></span> given by
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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} )}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
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<mi mathvariant="bold">x</mi>
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<mo stretchy="false">)</mo>
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<mn>1</mn>
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<mi>π<!-- π --></mi>
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<mo>∫<!-- ∫ --></mo>
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<mi>S</mi>
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<mi>ρ<!-- ρ --></mi>
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<mrow class="MJX-TeXAtom-ORD">
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ν<!-- ν --></mi>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
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<mo stretchy="false">|</mo>
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<mi>d</mi>
<mi>σ<!-- σ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle 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} )}</annotation>
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</math></span></span>
where <i>ρ</i> denotes the dipole distribution, <i>∂</i>/<i>∂ν</i> denotes the directional derivative in the direction of the outward unit normal in the <i>y</i> variable, and dσ is the surface measure on <i>S</i>.
</p><p>More generally, a double layer potential is associated to a <a href="Hypersurface" title="Hypersurface">hypersurface</a> <i>S</i> in <i>n</i>-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> by means of
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(\mathbf {x} )=\int _{S}\rho (\mathbf {y} ){\frac {\partial }{\partial \nu }}P(\mathbf {x} -\mathbf {y} )\,d\sigma (\mathbf {y} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
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<mo stretchy="false">)</mo>
<mo>=</mo>
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<mo>∫<!-- ∫ --></mo>
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<mi>S</mi>
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<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
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<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ν<!-- ν --></mi>
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</mrow>
<mi>P</mi>
<mo stretchy="false">(</mo>
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<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
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<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>σ<!-- σ --></mi>
<mo stretchy="false">(</mo>
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<annotation encoding="application/x-tex">{\displaystyle u(\mathbf {x} )=\int _{S}\rho (\mathbf {y} ){\frac {\partial }{\partial \nu }}P(\mathbf {x} -\mathbf {y} )\,d\sigma (\mathbf {y} )}</annotation>
</semantics>
</math></span></span>
where <i>P</i>(<b>y</b>) is the <a href="Newtonian_kernel" class="mw-redirect" title="Newtonian kernel">Newtonian kernel</a> in <i>n</i> dimensions.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Single_layer_potential" class="mw-redirect" title="Single layer potential">Single layer potential</a></li>
<li><a href="Potential_theory" title="Potential theory">Potential theory</a></li>
<li><a href="Electrostatics" title="Electrostatics">Electrostatics</a></li>
<li><a href="Laplacian_of_the_indicator" title="Laplacian of the indicator">Laplacian of the indicator</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFCourantHilbert1962" class="citation cs2"><a href="Richard_Courant" title="Richard Courant">Courant, Richard</a>; <a href="David_Hilbert" title="David Hilbert">Hilbert, David</a> (1962), <i>Methods of Mathematical Physics, Volume II</i>, Wiley-Interscience</cite>.</li>
<li><cite id="CITEREFKellogg1953" class="citation cs2">Kellogg, O. D. (1953), <i>Foundations of potential theory</i>, New York: <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-60144-1</bdi></cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{citation}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span>.</li>
<li><cite id="CITEREFShishmarev2001" class="citation cs2">Shishmarev, I.A. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Double-layer_potential">"Double-layer potential"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite>.</li>
<li><cite id="CITEREFSolomentsev2001" class="citation cs2">Solomentsev, E.D. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Multi-pole_potential">"Multi-pole potential"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite>.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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