Green's theorem
/ GREENZ /
Walk once counterclockwise around the boundary of a flat region, adding up how much a vector field pushes you along as you go — that is the circulation around the loop. Green's theorem makes a surprising claim: that boundary total equals a sum of the field's local swirling, the curl, gathered over the entire interior. The behavior on the edge is completely determined by what happens in the bulk, and vice versa.
Precisely, for a vector field F = (P, Q) on a region D in the plane with positively oriented boundary curve C, Green's theorem says the line integral over C of P dx + Q dy equals the double integral over D of (partial Q/partial x - partial P/partial y) dA. The integrand on the right is exactly the (scalar) curl of F in the plane. Read the other way — using P and Q swapped into a divergence — the same theorem says the flux of F out through the boundary equals the double integral of div F over the region. So Green's theorem has both a circulation form and a flux form, the two-dimensional shadows of Stokes' theorem and the divergence theorem.
Green's theorem is the first of the three great theorems and the template for all of them: it trades a harder boundary integral for an easier area integral (or the reverse). A famous consequence is the planimeter formula for area, Area = (1/2) line integral over C of (x dy - y dx), which computes a region's area purely from a trip around its edge — the principle behind the mechanical planimeter that surveyors once rolled around map outlines. It is also the workhorse for two-dimensional fluid flow and for shortcutting plane line integrals.
For F = (-y, x) around the unit circle, the line integral is 2 pi (computed earlier). Green's theorem confirms it instantly: partial Q/partial x - partial P/partial y = 1 - (-1) = 2, and the double integral of 2 over the unit disk is 2 times pi = 2 pi.
Loop integral and area integral of the curl agree — exactly what Green's theorem promises.
Green's theorem needs the boundary oriented positively (region on your left, counterclockwise for the outer edge) and the field smooth throughout D; if F has a singularity inside the region — like the vortex field at the origin — you must exclude it with an inner boundary, and skipping that step is the most common mistake.