# Stokes' theorem

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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.