Let \(S\) be the hemisphere \(x^2+y^2+z^2=1\), \(z\ge0\), with upward normal, and \(\mathbf F=\langle y,-x,0\rangle\). Compute \(\iint_S(\nabla\times\mathbf F)\cdot d\mathbf S\).
By Stokes the flux depends only on the boundary, the unit circle; using the disk as the surface, \(\nabla\times\mathbf F=\langle0,0,-2\rangle\) gives \(-2\cdot\pi=-2\pi\).
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 \(S\).