Vector analysis formulation
In vector analysis Stokes' theorem is commonly written as
where ∇ × F is the curl of a vector field on , the vector dS is a vector normal to the surface element dS, the contour integral is over a closed, non-intersecting path C bounding the open, two-sided surface S. The direction of the vector dS is determined according to the right screw rule by the direction of integration along C.
Differential geometry formulation
In differential geometry the theorem is extended to integrals of exterior derivatives over oriented, compact, and differentiable manifolds of finite dimension. It can be written as , where is a singular cube, and is a differential form.