Stokes' theorem

Lifts Green's tangential form off the flat plane: a curve's circulation traded for curl integrated as flux through any curved surface it bounds.

Stokes' Theorem states $\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_S \operatorname{curl}\mathbf{F}\cdot d\mathbf{S}$ , where $C$ is the boundary of an oriented surface $S$ , with $C$ 's direction and $S$ 's normal linked by the right-hand rule. Green's Theorem is the special case where $S$ is a flat region $R$ in the $xy$ -plane, so $\mathbf{n}=\mathbf{k}$ everywhere and $\operatorname{curl}\mathbf{F}\cdot d\mathbf{S}$ collapses to exactly the $\mathbf{k}$ -component of curl, $(Q_x-P_y)\,dA$ .

Examples

For $\mathbf{F}(x,y,z)=(-y,\ x,\ 0)$ and $C$ the unit circle in the $xy$ -plane, spanned by the flat unit disk: $\oint_C \mathbf{F}\cdot d\mathbf{r} = 2\pi$ .
A fluid's vorticity is $\operatorname{curl}\mathbf{v}$ , and Stokes' Theorem says a circulation meter around any loop reads the total vorticity through any surface spanning that loop.