Multivariable Calculus

Green’s Theorem

Given $C:$

• Closed
• Simple
• Piecewise Smooth
• Positively-Oriented i.e. CCW
• $P$ and $Q$ are $C^1$

If all the above are satisfied, then Green’s Theorem can be applied.

$${\oint }_C\mathbf{F}\cdot d\mathbf{r}={\oint }_{C}Pdx+Qdy={\iint }_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA$$