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

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

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, 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 $\int _{c}d\omega =\int _{\partial c}\omega$, where $c$ is a singular cube, and $\omega$ is a differential form.