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Let $C$ be a positively oriented, piecewise-smooth, simple closed curve in the plane and let $D$ be the region bounded by $C$ ($C$ is sometimes denoted $\partial D$ in this case, as the boundary of $D$). If $P$ and $Q$ have continuous partial derivatives on an open region that contains $D$, then \[ \oint_{\partial D} Pdx + Qdy = \int_{D}\int_{} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \]
Requiring an open region to contain $D$ means that the derivatives will exist on the boundary. That's why that's included.
Green's theorem is simply a calculation of a rather special integral on a two-dimensional region $D$: \[ \oint_{\partial D} {\bf{F}} \cdot d{\bf{r}} = \oint_{\partial D} Pdx + Qdy = \int_{D}\int_{} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)dA \] It shows one way to handle the ``work problem'' $\int_{C} {\bf{F}} \cdot d{\bf{r}}$ when the field is not conservative. It can also be seen as a generalization of the Fundamental Theorem of Calculus to area integrals, in the sense that the integral defined on a region can be evaluated by considering only its boundary.