Let \(S\) be the disk \(x^2+y^2\le1\) in the plane \(z=0\), with normal \(\mathbf k\). Compute \(\iint_S(\nabla\times\mathbf F)\cdot d\mathbf S\) for \(\mathbf F=\langle -y,x,0\rangle\).
\(\nabla\times\mathbf F=\langle0,0,2\rangle\) and its dot product with \(\mathbf k\) is \(2\); the flux is \(2\) times the area \(\pi\): \(2\pi\). It is the circulation of \(\mathbf F\) around the unit circle.
Stokes' theorem relates the circulation of a field \(\mathbf F\) along the closed curve \(C\) bounding an oriented surface \(S\) to the flux of the curl of \(\mathbf F\) through \(S\).