Stokes' theorem
/ STOHKS /
Take Green's theorem and lift it out of the flat plane into three-dimensional space, where the loop bounds not a flat patch but a bent surface stretched across it like a soap film on a wire ring. Stokes' theorem says the circulation of a vector field around the rim equals the total curl of the field passing through any surface that spans the rim. The swirl along the edge is the swirl gathered over the whole membrane.
Precisely, if S is an oriented surface with boundary curve C, then the line integral over C of F dot dr equals the flux integral over S of (curl F) dot n dS. The left side is circulation around the boundary; the right side is the flux of the curl through the surface. A striking feature is that the answer does not depend on which spanning surface you choose — any two surfaces with the same boundary give the same flux of curl, because the difference between them is a closed surface and div(curl F) = 0. Orientation must be consistent: the boundary is traversed by the right-hand rule relative to the surface normal.
Stokes' theorem is the engine of electromagnetism's circulation laws. Faraday's law and Ampere's law are usually written in their curl (differential) form, but their integral form — the EMF around a loop equals the rate of change of magnetic flux through it, the magnetic circulation around a loop equals the enclosed current — is exactly Stokes' theorem applied to curl E and curl B. It is also the conceptual reason a curl-free field on a simply connected region is conservative: every loop bounds a surface, and zero curl makes every circulation vanish.
Let F = (-y, x, 0) and let C be the unit circle in the xy-plane bounding the flat disk S. The circulation is 2 pi (as before); Stokes' theorem agrees, since curl F = (0, 0, 2), the upward normal is (0, 0, 1), so the flux of curl is 2 times the disk's area pi = 2 pi.
Circulation round the rim equals the flux of curl through any spanning surface — here, the flat disk.
The surface and its boundary must be oriented compatibly by the right-hand rule, and the field must be smooth on the surface; choosing the wrong orientation flips the sign, and a field with a singularity threaded through the loop (so no smooth spanning surface avoids it) can make the naive theorem fail.