Green’s Theorem

Conditions

1. $D$ has a smooth boundary $\partial D$ oriented positively
   • Outer boundaries counterclockwise
   • All inner boundaries clockwise
2. $\vec{F}$ has continuous partial derivatives in $D+\partial D$
3. $D$ is simply connected
   • If not vertically or horizontally simple, you must cut up

Statement

$${\oint }_{\partial D}\vec{F}\cdot d\vec{r}={\iint }_D\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dA$$

Usage

Area as a Line Integral

$$A(D)={\iint }_{D}dA$$

You can use various vector fields (depending on what’s easiest to solve)

• $\vec{F}=⟨0,x⟩⟶={\oint }_{\partial D}xdy$
• $\vec{F}=⟨-y,0⟩⟶=-{\oint }_{\partial D}ydx$
• $\vec{F}=⟨-\frac{1}{2}y,-\frac{1}{2}x⟩⟶=\frac{1}{2}{\oint }_{\partial D}xdy-ydx$

Contour Transformation Law

• $C$ and $C^{\prime }$ go $A\to B$
• $\partial D=C\cup -C^{\prime }$

$$\begin{array}{rl}& {\int }_C\vec{F}\cdot d\vec{r}-{\int }_{C\prime }\vec{F}\cdot d\vec{r}={\oint }_{\partial D}\vec{F}\cdot d\vec{r}\\ & \\ & {\int }_C\vec{F}\cdot d\vec{r}={\int }_{C\prime }\vec{F}\cdot d\vec{r}+{\iint }_D\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dA\end{array}$$