= Circulation Integral =

A '''circulation integral''' measures rotation.

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== Description ==

Circulation of a [[Calculus/VectorField|vector field]] ''F'' along a closed curve ''C'' is measured with a [[Calculus/LineIntegral|line integral]].

{{attachment:circ1.svg}}

where ''t̂'' is the unit tangent vector. See [[Calculus/LineIntegral|here]] for an explanation of ''dr''.



=== Green's Theorem ===

When the vector field is given as ''F = <P(x,y), Q(x,y)>'', '''Green's theorem''' gives a method for evaluating this.

{{attachment:circ2.svg}}

This can also be easily reformulated into a vector form that uses [[Calculus/Curl|curl]].

{{attachment:circ3.svg}}

where ''k̂'' is the [[Calculus/UnitVector|unit basis vector]].

Note the closely related [[Calculus/FluxIntegral#Closed_Line_Integrals|normal form of the theorem]].



=== Stoke's Theorem ===

For a surface ''S'' that is bounded by a closed curve ''C'', and given a vector field as  ''F = <P(x,y), Q(x,y)>'',, '''Stoke's theorem''' gives a method for evaluating this.

{{attachment:circ4.svg}}

where ''n̂'' is the [[Calculus/Orthogonality#Normal_Vectors|unit normal vector]]. It should be clear then that Green's theorem is a special case of Stoke's theorem, wherein the vector field is constrained to the ''xy''-plane, and therefore ''k̂'' is always the unit normal vector.

Note that, in some texts, the expression ''n̂ dS'' is rewritten as ''d'''S''''' or ''dS⃗''. This term represents the differential surface area vector.

An important application of this theorem is that such integrals are surface independent; they depend only on the curve ''C'' and the orientation of the surface ''S''. For two surfaces ''S,,1,,'' and ''S,,2,,'', as long as they have equivalent orientation and both follow ''C'', it must be that:

{{attachment:stokes.svg}}



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CategoryRicottone