Green's Theorem: Circulation and Curl

Green's theorem ties how much a field circulates around a closed loop to how much it swirls inside: $\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_A (\nabla\times\mathbf{F})\,dA$ in the plane, where the scalar curl $\nabla\times\mathbf{F} = \partial F_y/\partial x - \partial F_x/\partial y$ measures the local rate of rotation. The left side adds up the tangential push of the field all the way around the boundary; the right side adds up the curl, the tiny paddlewheel spin, over the whole interior. Drag the circle through a vector field and the two numbers stay locked together: the circulation around any loop equals the curl enclosed. This is the planar version of Stokes's theorem and the mathematics behind Ampere's law, where the field is the magnetic field and the curl is the current density, so the circulation of $\mathbf{B}$ around a loop counts the current threading it. The point-vortex field makes that exact: its circulation is the same constant whenever the vortex is enclosed and zero when it is not.