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''of a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022330/c0223301.png" /> along a closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022330/c0223302.png" />''
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''of a vector field  $  \mathbf a ( \mathbf r) $
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along a closed curve $  L $''
  
 
The integral
 
The integral
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022330/c0223303.png" /></td> </tr></table>
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$$
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\oint _ { L } \mathbf a  d \mathbf r .
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$$
  
 
In coordinate form the circulation is equal to
 
In coordinate form the circulation is equal to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022330/c0223304.png" /></td> </tr></table>
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$$
 
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\int\limits _ { L }
The work performed by the forces of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022330/c0223305.png" /> in displacing a test body (of unit mass, charge, etc.) along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022330/c0223306.png" /> is equal to the circulation of the field along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022330/c0223307.png" />. See [[Stokes theorem|Stokes theorem]].
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( a _ {x}  dx + a _ {y}  dy + a _ {z}  dz).
 
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$$
  
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The work performed by the forces of the field  $  \mathbf a ( \mathbf r ) $
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in displacing a test body (of unit mass, charge, etc.) along  $  L $
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is equal to the circulation of the field along  $  L $.
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See [[Stokes theorem|Stokes theorem]].
  
 
====Comments====
 
====Comments====
 
This notion is also called a line integral (of a vector field) along a closed curve.
 
This notion is also called a line integral (of a vector field) along a closed curve.

Latest revision as of 16:44, 4 June 2020


of a vector field $ \mathbf a ( \mathbf r) $ along a closed curve $ L $

The integral

$$ \oint _ { L } \mathbf a d \mathbf r . $$

In coordinate form the circulation is equal to

$$ \int\limits _ { L } ( a _ {x} dx + a _ {y} dy + a _ {z} dz). $$

The work performed by the forces of the field $ \mathbf a ( \mathbf r ) $ in displacing a test body (of unit mass, charge, etc.) along $ L $ is equal to the circulation of the field along $ L $. See Stokes theorem.

Comments

This notion is also called a line integral (of a vector field) along a closed curve.

How to Cite This Entry:
Circulation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Circulation&oldid=13781
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article