Green's Theorem

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Green's Theorem is a result in real analysis. It is a special case of Stokes' Theorem.

Statement

Let $D$ be a bounded subset of $\mathbb{R}^2$ with positively oriented boundary $C$ , and let $L$ and $M$ be functions with continuous partial derivatives mapping an open set containing $D$ into $\mathbb{R}$ . Then $\int_C \bigl[ L(x,y)dx + M(x,y)dy \bigr] = \iint_D \left(\frac{\partial M} {\partial x} - \frac{\partial L}{\partial y}\right...$

Proof

It suffices to show that the theorem holds when $D$ is a square, since $D$ can always be approximated arbitrarily well with a finite collection of squares.

Then let $D$ be a square with vertices $(a,c)$ , $(b,c)$ , $(b,d)$ , $(a,d)$ , with $a\le b$ and $c \le d$ . Then $\begin{align*}\int_C \bigl[ L(x,y)dx + M(x,y)dy \bigr] ={}&\int_{a}^b L(x,c) dx + \int_{c}^d M(b,y)dy \\&+ \int_b^a L...$ Now, by the Fundamental Theorem of Calculus, $L(x,d) - L(x,c) = \int_c^d \frac{\partial L}{\partial y}(x) dy$ and $M(b,y) - M(a,y) = \int_a^b \frac{\partial M}{\partial x}(y) dx .$ Hence $\begin{align*}\int_c^d \bigl[ M(b,y) - M(a,y) \bigr] dx - \int_a^b \bigl[ L(x,d) - L(x,c) \bigr] dy&= \int_c^d \int_a^b \...$ as desired. $\blacksquare$

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Green's Theorem

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Statement

Proof

See also