## Green's Theorem for Multiply Connected Regions

Green's Theorem normally stated applies to simply connected regions, so that any closed curve drawn in the region can be shrunk to a point. The region shown below is not simply connected.The area bounded by the closed curve DEF has been removed. By drawing a line AD from the boundary ABC to the boundary DEF, we can apply Green's the rem to the simply connected region ABCDEF, obtaining

\[\begin{equation} \begin{aligned} \int \int_R ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} dx \: dy &= \oint_{ABCDEF} P dx + Q dy \\ &= \int_{ABCA} P dx + Q dy + \int_{AD} P dx + Q dy + \oint_{DEFD} P dx + Q dy + \oint_{DA} P dx + Q dy \\ &= \int_{ABCA} P dx + Q dy + \oint_{DEFD} P dx + Q dy \\ &= \oint_{C} P dx + Q dy \end{aligned} \end{equation}\]

Where

\[C\]

is the boundary \[ABC \cup DEF\]

of the region \[R\]