Green's Theorem

The hypotheses matter: \(C\) must be closed, simple and piecewise smooth, traversed with the region \(D\) always on the left (counterclockwise), and \(P\), \(Q\) must have continuous partial derivatives on an open set containing \(D\). Traversing \(C\) backwards flips the sign. If \(D\) has holes, the full boundary includes the inner curves traversed clockwise.

The integrand \(Q_x-P_y\) is the \(\mathbf k\)-component of the curl of \(\langle P,Q,0\rangle\): the total circulation along the boundary is the sum of the small local circulations inside \(D\), because contributions along shared interior edges cancel in pairs. A guiding example is \(\langle -y,x\rangle\), with \(Q_x-P_y=2\): its circulation around any simple counterclockwise curve is twice the enclosed area.

This yields area formulas, such as \(A=\oint_C x\,dy=-\oint_C y\,dx=\tfrac12\oint_C x\,dy-y\,dx\), and a quick way to evaluate hard line integrals: if \(Q_x-P_y\) is simple, integrate over the region instead. Typical mistakes: forgetting the orientation, or applying the theorem to an open curve, where it does not hold.

If \(Q_x=P_y\) on a region without holes, the circulation around every closed curve is zero, which is the underlying reason a field with zero curl is conservative there. The theorem generalizes in two directions: Stokes' theorem, to surfaces in space, and the divergence theorem, to flux across a boundary.