Mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by , defined as the curl of the velocity field  describing the continuum motion. In Cartesian coordinates:
In words, the vorticity tells how the velocity vector changes when one moves by an infinitesimal distance in a direction perpendicular to it.
In a two-dimensional flow where the velocity is independent of the -coordinate and has no -component, the vorticity vector is always parallel to the -axis, and therefore can be expressed as a scalar field multiplied by a constant unit vector :
The vorticity is also related to the flows circulation (line integral of the velocity) along a closed path by the (classical) Stokes theorem. Namely, for any infinitesimal surface element  with normal direction  and area , the circulation  along the perimeter of  is the dot product  where  is the vorticity at the center of .