Stokes' Theorem

_C F· d r = _S ( × F)· d S — circulation around the boundary equals curl flux through any surface.

The idea

Theorem (Stokes' theorem).

Let $S$ be an oriented smooth surface in $\mathbb{R}^3$ whose boundary is a simple closed curve $C$, oriented compatibly with $S$. For every smooth vector field $\mathbf{F}$ on $S$, $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}.$

The left side is the circulation of $\mathbf{F}$ once around the rim: the push of the field along the direction of travel, totalled over the loop. The right side integrates $\nabla \times \mathbf{F}$ — the field's local rotation — over the whole surface the rim bounds.

The mechanism is cancellation, as in Green's theorem. Tile $S$ with small patches and traverse the boundary of each one. Every interior edge belongs to two patches, which traverse it in opposite directions, so its two contributions cancel in the sum. The edges that survive are those with no neighbouring patch, and together they make up the rim $C$. Each patch contributes, per unit of area, the component of $\nabla \times \mathbf{F}$ along its normal, so the surviving total is the integral on the right. Nothing in this argument used the shape of $S$: any surface with the same rim $C$ carries the same flux.

Compatible orientations keep the signs consistent — curl the right hand's fingers along the direction of travel around $C$, and the thumb points along the normal $\mathbf{n}$ on $S$. Reverse one without the other, and the two sides differ by a sign.

Ways to work on it

• Walkthrough. Trade a line integral around a loop for a surface integral of curl, and verify the two sides match.
• Practice. Compute a curl component for various 3D vector fields.
• Hardest. Compute a line integral around a closed loop via Stokes' theorem.

Not sure where to start? Take the ten-question placement test.