# Definition Let $S$ be an oriented [[smoothness|smooth]] surface and $\partial S$ its [[simple contour|simple]], closed, [[smoothness|smooth]] boundary curve with positive orientation. Let $\pmb{\Omega}$ be a vector field. Stokes' theorem states that: $ \int_S \int (\nabla \times \pmb{\Omega}) \cdot \text{d} \pmb{A} = \oint_{\partial S} \pmb{\Omega} \cdot \text{d} \pmb{r} $ The right-hand side is the [[flux]] of the [[curl]] of the vector field. It is also known as the fundamental theorem of curls or the curl theorem. This theorem is valid for vector calculus on $\mathbb{R}^3$. # Generalization The generalized Stokes' theorem states that the integral of a [[differential form]] $\Omega$ over the boundary of some [[Orientability|orientable]] [[manifold]] $S$ is equal to the integral of its exterior derivative $d\Omega$ over the whole of $S$. $ \int_{\partial S} \Omega = \int_{S} d\Omega $ In the $\mathbb{R}^3$ case it relates a [[one-form|one-form]] (the vector field) to a two-form (its curl, the exterior derivative).