Divergence theorem: Difference between revisions

Revision as of 14:04, 20 February 2012

divergence theorem

(Also called Gauss's theorem.) The statement that the volume integral of the divergence of a vector, such as the velocity V, over a volume V is equal to the surface integral of the normal component of V over the surface s of the volume (often called the "export" through the closed surface), provided that V and its derivatives are continuous and single-valued throughout V and s.

This may be written

where n is a unit vector normal to the element of surface ds, and the symbol ∮ ∮_S indicates that the integration is to be carried out over a closed surface. This theorem is sometimes called Green's theorem in the plane for the case of two-dimensional flow, and Green's theorem in space for the three-dimensional case described above. The divergence theorem is used extensively in manipulating the meteorological equations of motion.

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-<div class="definition"><div class="short_definition">(<br/>''Also called'' Gauss's theorem.) The statement that the volume integral of the  [[divergence]] of a [[vector]], such as the [[velocity]] '''V''', over a volume ''V'' is equal to the [[surface integral]]  of the normal component of '''V''' over the surface ''s'' of the volume (often called the &ldquo;export&rdquo; through  the closed surface), provided that '''V''' and its derivatives are continuous and single-valued throughout  ''V'' and ''s''.</div><br/> <div class="paragraph">This may be written  <div class="display-formula"><blockquote>[[File:ams2001glos-De37.gif|link=|center|ams2001glos-De37]]</blockquote></div> where '''n''' is a unit [[vector]] normal to the element of surface ''ds'', and the symbol &#x0222e; &#x0222e;<sub>''S''</sub> indicates that  the integration is to be carried out over a closed surface. This theorem is sometimes called [[Green's  theorem]] in the plane for the case of two-dimensional flow, and Green's theorem in space for the  three-dimensional case described above. The divergence theorem is used extensively in manipulating  the meteorological [[equations of motion]].</div><br/> </div>
+<div class="definition"><div class="short_definition">(''Also called'' Gauss's theorem.) The statement that the volume integral of the  [[divergence]] of a [[vector]], such as the [[velocity]] '''V''', over a volume ''V'' is equal to the [[surface integral]]  of the normal component of '''V''' over the surface ''s'' of the volume (often called the "export" through  the closed surface), provided that '''V''' and its derivatives are continuous and single-valued throughout  ''V'' and ''s''.</div><br/> <div class="paragraph">This may be written  <div class="display-formula"><blockquote>[[File:ams2001glos-De37.gif|link=|center|ams2001glos-De37]]</blockquote></div> where '''n''' is a unit [[vector]] normal to the element of surface ''ds'', and the symbol &#x0222e; &#x0222e;<sub>''S''</sub> indicates that  the integration is to be carried out over a closed surface. This theorem is sometimes called [[Green's  theorem]] in the plane for the case of two-dimensional flow, and Green's theorem in space for the  three-dimensional case described above. The divergence theorem is used extensively in manipulating  the meteorological [[equations of motion]].</div><br/> </div>
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