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\title{Proof of Green's Theorem}
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\date{}
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\begin{theorem}[Green's Theorem]
Let \( C \) be a positively oriented, simple closed curve in the plane, and let \( D \) be the region bounded by \( C \). If \( L(x, y) \) and \( M(x, y) \) have continuous partial derivatives on an open region that contains \( D \) and \( C \), then
\[
\oint_C \left( L \, dx + M \, dy \right) = \iint_D \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dA.
\]
\end{theorem}

\begin{proof}
We will prove Green's Theorem by breaking the region \( D \) into small rectangles and then using the Fundamental Theorem of Calculus.

Assume that the region \( D \) is divided into \( m \times n \) small rectangles. For each small rectangle \( R_{ij} \) with vertices \((x_i, y_j)\), \((x_{i+1}, y_j)\), \((x_{i+1}, y_{j+1})\), and \((x_i, y_{j+1})\), we approximate the line integral around the boundary of \( R_{ij} \):

\[
\oint_{\partial R_{ij}} \left( L \, dx + M \, dy \right) \approx \left( M(x_{i+1}, y_{j+1}) - M(x_{i}, y_{j+1}) \right)(x_{i+1} - x_i) - \left( L(x_{i+1}, y_{j+1}) - L(x_{i+1}, y_j) \right)(y_{j+1} - y_j).
\]

This expression can be rewritten as:

\[
\oint_{\partial R_{ij}} \left( L \, dx + M \, dy \right) \approx \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) \Delta x \Delta y,
\]
where \( \Delta x = x_{i+1} - x_i \) and \( \Delta y = y_{j+1} - y_j \).

Summing over all rectangles in the region \( D \), we obtain:

\[
\sum_{i,j} \oint_{\partial R_{ij}} \left( L \, dx + M \, dy \right) \approx \sum_{i,j} \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) \Delta x \Delta y.
\]

The left-hand side of this equation is approximately the line integral over \( C \), and the right-hand side is a Riemann sum that approximates the double integral over \( D \):

\[
\oint_C \left( L \, dx + M \, dy \right) = \iint_D \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dA.
\]

Thus, Green's Theorem is proved.
\end{proof}

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