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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  
\[R\]