A special case of Stokes' theorem in which F is a vector field and M is an oriented, compact embedded 2-manifold with boundary in R^3, and a generalization of Green's theorem from the plane into three-dimensional space. The curl theorem states integral_S( del xF)·d a = integral_(dS) F·d s, where the left side is a surface integral and the right side is a line integral. There are also alternate forms of the theorem. If F congruent c F, then integral_S d ax del F = integral_C F d s.

change of variables theorem | curl | divergence theorem | Green's theorem | Stokes' theorem