Stokes' theorem

Statement

Let $D \subseteq \mathbb{R}^2$ be the compact region bounded by a piecewise smooth simple closed curve $\gamma$, oriented counterclockwise. Let $r$ be an injective regular parametrised surface of class $C^2$ on an open set that contains $D$. Let $S = r(D)$ have the unit normal $n = (r_u \times r_v) / \| r_u \times r_v \|$, and let $\partial S$ be the closed curve $r \circ \gamma$. If $F$ is a vector field of class $C^1$ on an open set that contains $S$, then $$\oint_{\partial S} F \cdot d\mathbf{r} = \iint_S (\nabla \times F) \cdot n \, dS.$$

Remarks

The left side is the circulation of $F$ around the boundary. The right side is the flux of the curl of $F$ through $S$. The orientations match by the right-hand rule: with the thumb along $n$, the fingers curl in the direction of $\partial S$. Example: for the upper unit hemisphere and $F = (-y, x, 0)$, $\nabla \times F = (0, 0, 2)$, and both sides equal $2\pi$. For a flat surface in the $xy$-plane, the theorem reduces to Green's theorem. The proof pulls the integrals back to $D$ and applies Green's theorem there.

History

The theorem first appears in a letter from William Thomson to George Stokes in 1850. Stokes set it as a question on the Smith's Prize examination at Cambridge in 1854, and the result then got his name.

Explanations

L1 human

The total swirl of a field over a surface equals the circulation of the field around the edge of the surface. The surface can be any shape: only its boundary curve matters.