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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Green's Theorem">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 30 14:20:50 1997}
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\section{Green's Theorem\label{Green's Theorem}}

\begin{itemize}
\item  Let $D$ be a region in the plane bounded by a simple closed curve $C$%
, which is specified by the vector function $\mathbf{r}(t)$ for $a\leq t\leq
b$. \ Assume that $D$ consists of all points inside $C$ as well as all the
points of $C$. \ Also assume that $C$ is \textbf{positively oriented}, which
means that as you traverse $C$ in the direction of increasing $t$, the
region $D$ is always on your left. \ 
\end{itemize}

\hfill

\begin{theorem}[\vspace{1pt}Green's Theorem]
Let $C$ be a positively oriented, piecewise-smooth, simple closed curve in
the plane and let $D$ be the region bounded by $C$. \ If $P$ and $Q$ have
continuous partial derivatives on an open region containing $D$, then 
\[
\int_{C}Pdx+Qdy=\iint_{D}\left( \frac{\partial Q}{\partial x}-\frac{\partial
P}{\partial y}\right) dA
\]
\end{theorem}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Iterated integrals}{}{}{../../Help/DM7-8.tex#Iterated
integrals}}

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