Stokes’ Theorem

Stokes’ Theorem:

$\oint_{\partial\Sigma} \vec{F} \cdot d\vec{l} = \iint_{\Sigma} (\nabla \times \vec{F}) \cdot d\vec{A}$

In Vector Calculus, Stokes’ Theorem gives equivalence between a closed contour (curve) and the surface it bounds.

Given a vector function $\vec{F}(x,y,z)$ , and a 2D region $\Sigma$ (with $\partial\Sigma$ representing the boundary of $\Sigma$ ),

The integral of $\vec{F}$ over $\partial\Sigma$ is equal to the integral of the curl of $\vec{F} (\nabla \cdot \vec{F})$ over $\Sigma$ .