## Proof of Green's Extended Theorem

Green's Extended Theorem in the plane applies to any region such that a line parallel to one of the coordinate axes cuts the curve in more than to places. We can prove Greens Extended Theorem by simply cutting the curve and it's interior into a set of curves, on each of which Green's Theorem can apply since the region is concave.. The union of the set of curves is the whole problematic curve. adding the integrals proves Green's Extended Theorem. This is illustrated for a region split into 4, with four boundary curves.For each of the regions

\[R_1 , R_2 , ..., R_n\]

, \[ \oint_{C_i} P \: dx + Q \: dy = \int \int_{R_i} \frac{ \partial Q}{\partial x} - \frac{ \partial P}{\partial y} dx \: dy \]

We can write Down a corresponding equation for each of

\[i=1,2,...,n\]

Adding them all gives

\[\begin{equation} \begin{aligned} \oint_{C_1} P \: dx + Q \: dy + \oint_{C_2} P \: dx + Q \: dy +...+ \oint_{C_n} P \: dx + Q \: dy &= \int \int_{R_1} \frac{ \partial Q}{\partial x} - \frac{ \partial P}{\partial y} dx \: dy + \int \int_{R_2} \frac{ \partial Q}{\partial x} - \frac{ \partial P}{\partial y} dx \: dy +... \\ &+ \int \int_{R_n} \frac{ \partial Q}{\partial x} - \frac{ \partial P}{\partial y} dx \: dy \end{aligned} \end{equation} \]

Which simplifies by the properties of integration to

\[\oint_{C} P \: dx + Q \: dy = \int \int_{R} \frac{ \partial Q}{\partial x} - \frac{ \partial P}{\partial y} dx \: dy \]