Editing Circulation (physics) (section)

==Definition and properties==

If {{math|'''V'''}} is a vector field and {{math|d'''l'''}} is a vector representing the [[Differential (infinitesimal)|differential]] length of a small element of a defined curve, the contribution of that differential length to circulation is {{math|dΓ}}:
<math display="block">\mathrm{d}\Gamma = \mathbf{V} \cdot \mathrm{d}\mathbf{l} = \left|\mathbf{V}\right| \left|\mathrm{d}\mathbf{l}\right| \cos \theta.</math>

Here, {{math|''θ''}} is the angle between the vectors {{math|'''V'''}} and {{math|d'''l'''}}.

The '''circulation''' {{math|Γ}} of a vector field {{math|'''V'''}} around a [[closed curve]] {{math|''C''}} is the [[line integral]]:<ref>{{cite book 
  | title = Introduction to Fluid Mechanics 
  | author1 =  Robert W. Fox 
  | author2 =  Alan T. McDonald
  | author3 =  Philip J. Pritchard 
  | edition = 6
  | publisher = [[John Wiley & Sons|Wiley]]
  | year = 2003
  | isbn = 978-0-471-20231-8
  }}</ref><ref name=":0">{{Cite web| title=The Feynman Lectures on Physics Vol. II Ch. 3: Vector Integral Calculus| url=https://feynmanlectures.caltech.edu/II_03.html|access-date=2020-11-02 | website=feynmanlectures.caltech.edu}}</ref>
<math display="block">\Gamma = \oint_{C}\mathbf{V} \cdot \mathrm d \mathbf{l}.</math>

In a [[conservative vector field]] this integral evaluates to zero for every closed curve. That means that a line integral between any two points in the field is independent of the path taken. It also implies that the vector field can be expressed as the [[gradient]] of a scalar function, which is called a [[Scalar potential|potential]].<ref name=":0" />
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The dimensions of circulation in fluid dynamics are length squared, divided by time; L2⋅T−1, which is equivalent to velocity times length.
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