Difference between revisions of "Circulation"
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| + | ''of a vector field $ \mathbf a ( \mathbf r) $ | ||
| + | along a closed curve $ L $'' | ||
The integral | The integral | ||
| − | + | $$ | |
| + | \oint _ { L } \mathbf a d \mathbf r . | ||
| + | $$ | ||
In coordinate form the circulation is equal to | 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|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=46347
Circulation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Circulation&oldid=46347
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article