# Stokes' theorem  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

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 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.