Green's Theorem

Green’s Theorem equates

• An area of a fence (Line Integral)
• A volume under a surface (Double Integral) Let $\vec{F}=⟨P,Q⟩$

$${\oint }_{C}Pdx+Qdy=\oint \vec{F}\cdot d\vec{r}={\iint }_D{Q}_x-P_{y}dA$$

$C$ must be positively-oriented, meaning that $D$ is on the left hand side when walking the curve. This is simply just because if it were negative oriented, the sign of the Line Integral would flip.