Let \(\mathbf F=\langle z,x,y\rangle\) and \(C\) the boundary of the triangle of the plane \(x+y+z=1\) in the first octant, counterclockwise seen from above. Compute \(\oint_C\mathbf F\cdot d\mathbf r\).
\(\nabla\times\mathbf F=\langle1,1,1\rangle\). With \(z=1-x-y\), \(d\mathbf S=\langle1,1,1\rangle dA\), the integrand is \(3\) and the projected area is \(\tfrac12\): \(\tfrac32\).
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 \(S\).