In the Euclidean space \(\mathbb{R}^3\) the plane \(z=\frac{1}{2}\) cuts the unit sphere \(S^2=\big\{{(x,y,z)\in\mathbb{R}^3\;\; x^2+y^2+z^2=1}\big\}\) in two surfaces \(S\) and \(T\). Let \(S\) be the surface which passes through the point \((0,0,1)\). Let the vector field \[\overline{F}(x,y,z)=\Bigg({\begin{array}{c} xy\\ yz\\ zx \end{array}}\Bigg)\,,\quad (x,y,z)\in\mathbb{R}^3\,.\] Verify Stokes' theorem for the field \(\overline{F}\) and the surface \(S\).
_________________ Grigorios Kostakos
