Green’s Theorem

Orientation of boundary

Let \( C=\partial D \) be the boundary of a closed and bounded region \( D \) in \( \mathbb{R}^{2} \). SUppose \( \partial D \) is the union of finitely many simple closed curves. We say that \( \partial D \) is positively oriented if \( D \) always lies on the left when one transverses \( \partial D \). Otherwise, we say \( \partial D \) is negatively oriented.

Green’s Theorem. Let \( \partial D \) be the boundary of a closed and bounded region \( D \) in \( \mathbb{R}^{n} \). Suppose \( \partial D \) is positively oriented and is the union of finitely many simple closed \( C^{1} \) curves. Let \( \mathbf{F}:X\mapsto\mathbb{R}^{2} \) be a vector field of class \( C^{1} \) where \( X\subset\mathbb{R}^{2} \) contains \( D \). Then

\begin{equation} \label{org25f491c} \oint_{\partial D} \mathbf{F}\cdot d\mathbf{s} = \oint_{\partial D} F_{1}dx+F_{2}dy = \iint_{D}\left( \frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y} \right) \, dxdy \end{equation}