Circulation

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[edit]

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