Stokes' theorem

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

In vector analysis it is commonly written as

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

where ∇ × F is the curl of a vector field on $\scriptstyle \mathbb {R} ^{3}$, the vector dS is a vector normal to the surface element dS, the contour integral is over a closed path C bounding the surface S.

In differential geometry the theorem is extended to integrals of exterior derivatives over oriented, compact, and differentiable manifolds of finite dimension.