Editing Circulation (physics) (section)

== Relation to vorticity and curl ==

Circulation can be related to [[Curl (mathematics)|curl]] of a vector field {{math|'''V'''}} and, more specifically, to [[vorticity]] if the field is a fluid velocity field,
<math display="block">\boldsymbol{\omega} = \nabla\times\mathbf{V}.</math>

By [[Stokes' theorem]], the [[flux]] of curl or vorticity vectors through a surface ''S'' is equal to the circulation around its perimeter,<ref name=":0" />
<math display="block">\Gamma = \oint_{\partial S} \mathbf{V}\cdot \mathrm{d}\mathbf{l} = \iint_S \nabla \times \mathbf{V} \cdot \mathrm{d}\mathbf{S}=\iint_S \boldsymbol{\omega} \cdot \mathrm{d}\mathbf{S}</math>

Here, the closed integration path {{math|''∂S''}} is the [[boundary (topology)|boundary]] or perimeter of an open surface {{math|''S''}}, whose infinitesimal element [[Normal (geometry)|normal]] {{math|1=d'''S''' = '''n'''dS}} is oriented according to the [[Right-hand rule#Curve orientation and normal vectors|right-hand rule]]. Thus curl and vorticity are the circulation per unit area, taken around a local infinitesimal loop.

In [[potential flow]] of a fluid with a region of [[vorticity]], all closed curves that enclose the vorticity have the same value for circulation.<ref name="JDA">Anderson, John D. (1984), ''Fundamentals of Aerodynamics'', section 3.16. McGraw-Hill. {{ISBN|0-07-001656-9}}</ref>