Theorem (Green's Theorem): If $D$ is a simple region and $P$, $Q$ have continuous partial derivatives on an open set containing $D$, then $\oint_{\partial D} P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$

It suffices to prove separately that $\oint P\,dx = -\iint_D \frac{\partial P}{\partial y}\,dA$ and $\oint Q\,dy = \iint_D \frac{\partial Q}{\partial x}\,dA$.

Step 1: Prove $\oint_C P\,dx = -\iint_D \frac{\partial P}{\partial y}\,dA$.

Assume $D$ is a type I region: $D = \{(x,y) : a \leq x \leq b, \; g_1(x) \leq y \leq g_2(x)\}$. The boundary $\partial D$ consists of four pieces:

• $C_1$: bottom, $y = g_1(x)$ from $x = a$ to $x = b$
• $C_2$: right side, $x = b$ from $y = g_1(b)$ to $y = g_2(b)$
• $C_3$: top, $y = g_2(x)$ from $x = b$ to $x = a$
• $C_4$: left side, $x = a$ from $y = g_2(a)$ to $y = g_1(a)$

On $C_2$ and $C_4$, $x$ is constant so $dx = 0$, hence $\int_{C_2} P\,dx = \int_{C_4} P\,dx = 0$.

On $C_1$: parametrize by $x$ from $a$ to $b$: $\int_{C_1} P\,dx = \int_a^b P(x, g_1(x))\,dx$

On $C_3$: parametrize by $x$ from $b$ to $a$: $\int_{C_3} P\,dx = \int_b^a P(x, g_2(x))\,dx = -\int_a^b P(x, g_2(x))\,dx$

Therefore: $\oint_C P\,dx = \int_a^b [P(x, g_1(x)) - P(x, g_2(x))]\,dx$

For the double integral, using Fubini: $-\iint_D \frac{\partial P}{\partial y}\,dA = -\int_a^b \int_{g_1(x)}^{g_2(x)} \frac{\partial P}{\partial y}\,dy\,dx = -\int_a^b [P(x, g_2(x)) - P(x, g_1(x))]\,dx$

$= \int_a^b [P(x, g_1(x)) - P(x, g_2(x))]\,dx = \oint_C P\,dx \quad \checkmark$

Step 2: Prove $\oint_C Q\,dy = \iint_D \frac{\partial Q}{\partial x}\,dA$.

The argument is analogous, treating $D$ as a type II region $D = \{(x,y) : c \leq y \leq d,\; h_1(y) \leq x \leq h_2(y)\}$. On the portions where $y$ is constant, $dy = 0$. The remaining integrals give: $\oint_C Q\,dy = \int_c^d [Q(h_2(y), y) - Q(h_1(y), y)]\,dy = \int_c^d \int_{h_1(y)}^{h_2(y)} \frac{\partial Q}{\partial x}\,dx\,dy = \iint_D \frac{\partial Q}{\partial x}\,dA$

Step 3: Combine.

Adding the two results: $\oint_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA \quad \square$