Stokes' Theorem

Let surface $S$ be an oriented, piecewise-smooth surface bounded by a simple, closed, piecewise-smooth boundary curve $C$, that is positively oriented.

Let $\vec{F}$ be a vector field whose components have continuous partial derivatives on an open region in ${\mathbb{R}}^3$ that contains $S$

$${\int }_C\vec{F}\cdot d\vec{r}={\iint }_S(\nabla \times \vec{F})\cdot d\vec{S}$$

• Line Integral is equal to Vector Surface Integral of the Curl
• “Draw a loop around it and it tells you how much swirly is in it”