# Circulation

From Encyclopedia of Mathematics

*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

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