# 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

${\displaystyle \iint _{S}\,({\boldsymbol {\nabla }}\times \mathbf {F} )\cdot d\mathbf {S} =\oint _{C}\mathbf {F} \cdot d\mathbf {s} }$

where × F is the curl of a vector field on ${\displaystyle \scriptstyle \mathbb {R} ^{3}}$, 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 ${\displaystyle \int _{c}d\omega =\int _{\partial c}\omega }$, where ${\displaystyle c}$ is a singular cube, and ${\displaystyle \omega }$ is a differential form.