Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls, or simply the curl theorem, is a theorem in vector calculus on ${\displaystyle \mathbb {R} ^{3}}$. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence:

Stokes' theorem is a special case of the generalized Stokes theorem. In particular, a vector field on ${\displaystyle \mathbb {R} ^{3}}$ can be considered as a 1-form in which case its curl is its exterior derivative, a 2-form.

Let ${\displaystyle \Sigma }$ be a smooth oriented surface in ${\displaystyle \mathbb {R} ^{3}}$ with boundary ${\displaystyle \partial \Sigma \equiv \Gamma }$. If a vector field ${\displaystyle \mathbf {F} (x,y,z)=(F_{x}(x,y,z),F_{y}(x,y,z),F_{z}(x,y,z))}$ is defined and has continuous first order partial derivatives in a region containing ${\displaystyle \Sigma }$, then $${\displaystyle \iint _{\Sigma }(\nabla \times \mathbf {F} )\cdot \mathrm {d} \mathbf {\Sigma } =\oint _{\partial \Sigma }\mathbf {F} \cdot \mathrm {d} \mathbf {\Gamma } .}$$ More explicitly, with ${\displaystyle \wedge }$ being the wedge product, the equality says that $${\displaystyle {\begin{aligned}&\iint _{\Sigma }\left(\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right)\,\mathrm {d} y\wedge \mathrm {d} z+\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right)\,\mathrm {d} z\wedge \mathrm {d} x+\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)\,\mathrm {d} x\wedge \mathrm {d} y\right)\\&=\oint _{\partial \Sigma }{\Bigl (}F_{x}\,\mathrm {d} x+F_{y}\,\mathrm {d} y+F_{z}\,\mathrm {d} z{\Bigr )}.\end{aligned}}}$$

The main challenge in a precise statement of Stokes' theorem is in defining the notion of a boundary. Surfaces such as the Koch snowflake, for example, are well-known not to exhibit a Riemann-integrable boundary, and the notion of surface measure in Lebesgue theory cannot be defined for a non-Lipschitz surface. One (advanced) technique is to pass to a weak formulation and then apply the machinery of geometric measure theory; for that approach see the coarea formula. In this article, we instead use a more elementary definition, based on the fact that a boundary can be discerned for full-dimensional subsets of ${\displaystyle \mathbb {R} ^{2}}$.

A more detailed statement will be given for subsequent discussions. Let ${\displaystyle \gamma :[a,b]\to \mathbb {R} ^{2}}$ be a piecewise smooth Jordan plane curve: a simple closed curve in the plane. The Jordan curve theorem implies that ${\displaystyle \gamma }$ divides ${\displaystyle \mathbb {R} ^{2}}$ into two components, a compact one and another that is non-compact. Let ${\displaystyle D}$ denote the compact part; then ${\displaystyle D}$ is bounded by ${\displaystyle \gamma }$. It now suffices to transfer this notion of boundary along a continuous map to our surface in ${\displaystyle \mathbb {R} ^{3}}$. But we already have such a map: the parametrization of ${\displaystyle \Sigma }$.

Suppose ${\displaystyle \psi :D\to \mathbb {R} ^{3}}$ is piecewise smooth at the neighborhood of ${\displaystyle D}$　, with ${\displaystyle \Sigma =\psi (D)}$. If ${\displaystyle \Gamma }$ is the space curve defined by ${\displaystyle \Gamma (t)=\psi (\gamma (t))}$ then we call ${\displaystyle \Gamma }$ the boundary of ${\displaystyle \Sigma }$, written ${\displaystyle \partial \Sigma }$

With the above notation, if ${\displaystyle \mathbf {F} }$ is any smooth vector field on ${\displaystyle \mathbb {R} ^{3}}$, then$${\displaystyle \oint _{\partial \Sigma }\mathbf {F} \,\cdot \,\mathrm {d} {\mathbf {\Gamma } }=\iint _{\Sigma }\nabla \times \mathbf {F} \,\cdot \,\mathrm {d} \mathbf {\Sigma } .}$$

Here, the "${\displaystyle \cdot }$" represents the dot product in ${\displaystyle \mathbb {R} ^{3}}$.