From AMS Glossary
A vector operation upon a vector field that represents the rotation of the field, related by Stokes's theorem to the circulation of the field at each point.
The curl is invariant with respect to coordinate transformations and is usually written
where ∇ is the del operator. In Cartesian coordinates, if F has the components, Fx, Fy, Fz, the curl is
Expansions in other coordinate systems may be found in any text on vector analysis. The curl of the velocity vector is called the vorticity; in a field of solid rotation it is equal to twice the angular velocity. Occasionally the vorticity is defined as one-half the curl. The curl of a two- dimensional vector field is always normal to the vectors of the field; this is not necessarily true in the three dimensions.