Difference between revisions of "Green's theorem"

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<div class="definition"><div class="short_definition">A form of the [[divergence theorem]] applied to a [[vector field]] so chosen as to  yield a formula useful in applying the [[Green's function]] method of solution of a [[boundary-value  problem]].</div><br/> <div class="paragraph">The most common form of the theorem is  <div class="display-formula"><blockquote>[[File:ams2001glos-Ge47.gif|link=|center|ams2001glos-Ge47]]</blockquote></div> where ''dV'' and ''dS'' are elements of the volume ''V'' and closed bounding surface ''S'', respectively; &#x003c6;  and &#x003c8; are any twice differential functions with continuous second partial derivatives in ''V''; ''n'' is the  outer normal to ''S''; and &nabla;<sup>2</sup> is the [[Laplacian operator]].</div><br/> </div>
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<div class="definition"><div class="short_definition">A form of the [[divergence theorem]] applied to a [[vector field]] so chosen as to  yield a formula useful in applying the [[Green's function]] method of solution of a [[boundary-value problem|boundary-value  problem]].</div><br/> <div class="paragraph">The most common form of the theorem is  <div class="display-formula"><blockquote>[[File:ams2001glos-Ge47.gif|link=|center|ams2001glos-Ge47]]</blockquote></div> where ''dV'' and ''dS'' are elements of the volume ''V'' and closed bounding surface ''S'', respectively; &#x003c6;  and &#x003c8; are any twice differential functions with continuous second partial derivatives in ''V''; ''n'' is the  outer normal to ''S''; and &nabla;<sup>2</sup> is the [[Laplacian operator]].</div><br/> </div>
 
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Latest revision as of 17:05, 25 April 2012



Green's theorem

A form of the divergence theorem applied to a vector field so chosen as to yield a formula useful in applying the Green's function method of solution of a boundary-value problem.

The most common form of the theorem is
ams2001glos-Ge47
where dV and dS are elements of the volume V and closed bounding surface S, respectively; φ and ψ are any twice differential functions with continuous second partial derivatives in V; n is the outer normal to S; and ∇2 is the Laplacian operator.


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