Curl (mathematics) - Definition

Definition

The curl of a vector field F, denoted by curl F or ∇ × F, at a point is defined in terms of its projection onto various lines through the point. If is any unit vector, the projection of the curl of F onto is defined to be the limiting value of a closed line integral in a plane orthogonal to as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.

As such, the curl operator maps C1 functions from R3 to R3 to C0 functions from R3 to R3.

Implicitly, curl is defined by:

where is a line integral along the boundary of the area in question, and |A| is the magnitude of the area. If is an outward pointing in-plane normal, whereas is the unit vector perpendicular to the plane (see caption at right), then the orientation of C is chosen so that a tangent vector to C is positively oriented if and only if forms a positively oriented basis for R3 (right-hand rule).

The above formula means that the curl of a vector field is defined as the infinitesimal area density of the circulation of that field. To this definition fit naturally

• the Kelvin-Stokes theorem, as a global formula corresponding to the definition, and
• the following "easy to memorize" definition of the curl in curvilinear orthogonal coordinates, e.g. in cartesian coordinates, spherical, cylindrical, or even elliptical or parabolical coordinates:

If (x₁, x₂, x₃) are the Cartesian coordinates and (u₁,u₂,u₃) are the orthogonal coordinates, then

is the length of the coordinate vector corresponding to u_i. The remaining two components of curl result from cyclic permutation of indices: 3,1,2 → 1,2,3 → 2,3,1.