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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.


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:
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article