Green's Theorem

Green's theorem simply says that what happens on the edge of a region tells you what's going on inside it. On the left, $\oint_C P\,dx + Q\,dy$ is the circulation: how much the field pushes you along as you walk once around the boundary $C$. On the right, $Q_x - P_y$ is the little bit of spin the field has at each point inside (the same expression from the conservative-field test), and $\iint_D$ adds it up over the whole region $D$. The theorem says the two are equal. It looks like a mouthful, but it's one sentence: total spin inside equals circulation around the edge.

Why it's true. Chop $D$ into tiny squares. Walk counterclockwise around one of them, and the circulation you pick up is about (spin there) × (area of the square); that's essentially what spin means. Now add up the circulations of all the little squares. Think about an edge shared by two neighbors: you walk it one way for the left square and the opposite way for the right square, so those two contributions cancel exactly. Every interior edge cancels like that. All that survives is the outer boundary, walked once counterclockwise. So the sum of (spin × area) over the inside equals the circulation around the outside.

It's one more case of the same big idea as the Fundamental Theorem of Calculus: integrate a derivative over a region and you only need the boundary. For $\int_a^b F'\,dx$ the boundary is two points; here it's a curve.

Use it either way:

• A hard line integral around a closed curve can become an easy double integral.
• For a conservative field the spin is $0$ everywhere, so the circulation around any loop is $0$. That's why the cross-partial test works on regions without holes: with no holes, every loop encloses only points where the spin is zero.
• Area from a boundary: with $Q = x$, $P = 0$, the spin is 1, so $\text{Area} = \oint_C x\,dy = \frac12\oint_C x\,dy - y\,dx$. That's how a planimeter measures area by tracing a boundary.

A common mistake. Green's theorem assumes you go around counterclockwise, keeping the region on your left. Go clockwise and the line integral flips sign, so you'd get the negative of the double integral. It's also only for closed curves: for an open path, there's no region inside to integrate over.

Mechanical planimeters measure the area of a map region by tracing its boundary, and computer graphics compute polygon areas with the discrete form of Green's theorem (the shoelace formula).