Conservative vector field - Revision history

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2014-12-31T14:55:45Z

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en>Permafrost46: WPCleaner v1.30 - Repaired 1 link to disambiguation page - (You can help) - Closed form

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← Older revision		Revision as of 17:50, 13 January 2014
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	~~Hey! I am Dalton. Acting should be~~ a ~~thing that I~~'~~m totally addicted to~~. ~~Vermont has become where my home~~ is. ~~Managing people has been my 24~~-~~hour period job for a while but I plan on changing~~ it. I'~~ve been toiling on my website just for some time now. Check~~ it ~~out here~~: ~~http~~:/~~/prometeu~~.~~net~~<br><br>~~Here~~ is ~~my web blog [http~~:/~~/prometeu~~.~~net Hack Clash Of Clans~~]		{{Unreferenced\|date=November 2009}}
			In [[vector calculus]], a '''Laplacian vector field''' is a [[vector field]] which is both [[irrotational field\|irrotational]] and [[incompressible fluid\|incompressible]]. If the field is denoted as '''v''', then it is described by the following [[differential equations]]:
			:<math> \nabla \times \mathbf{v} = 0, </math>
			:<math> \nabla \cdot \mathbf{v} = 0. </math>
			A Laplacian vector field in the plane satisfies the [[Cauchy-Riemann equations]]: it is [[holomorphic]].

			Since the [[Curl (mathematics)\|curl]] of '''v''' is zero, it follows that '''v''' can be expressed as the gradient of a [[scalar potential]] (see [[irrotational field]]) ''φ'' :
			:<math> \mathbf{v} = \nabla \phi \qquad \qquad (1) </math>.
			Then, since the [[divergence]] of '''v''' is also zero, it follows from equation (1) that
			:<math> \nabla \cdot \nabla \phi = 0 </math>
			which is equivalent to
			:<math> \nabla^2 \phi = 0 </math>.
			Therefore, the potential of a Laplacian field satisfies [[Laplace's equation]].

			==See also==
			* [[potential flow]]
			* [[harmonic function]]

			[[Category:Vector calculus]]


			{{Mathanalysis-stub}}

en>Gabbe: Reverted edits by 14.98.22.59 (talk) to last version by 208.80.119.68

2012-08-21T08:27:18Z

Reverted edits by 14.98.22.59 (talk) to last version by 208.80.119.68

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Hey! I am Dalton. Acting should be a thing that I'm totally addicted to. Vermont has become where my home is. Managing people has been my 24-hour period job for a while but I plan on changing it. I've been toiling on my website just for some time now. Check it out here: http://prometeu.net<br><br>Here is my web blog [http://prometeu.net Hack Clash Of Clans]