# Stokes' theorem

In vector analysis and differential geometry, **Stokes' theorem** is a statement that treats integrations of differential forms.

## Vector analysis formulation

In vector analysis Stokes' theorem is commonly written as

where **∇** × **F** is the curl of a vector field on , the vector d**S**
is a vector normal to the surface element d*S*, the contour integral is over a closed, non-intersecting path *C* bounding the open, two-sided surface *S*. The direction of the vector d**S** 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.