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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 \[\overline{F}(x,y,z)=\Bigg({\begin{array}{c} x-y\\ y-z\\ z-x \end{array}}\Bigg)\,,\quad (x,y,z)\in\mathbb{R}^3\,.\] Verify Stokes' theorem for the field \(\overline{F}\) and the surface \(S\).