Verify Stokes' Theorem for \(\mathbf{F}(x, y, z)=\langle y, z-x, 0angle\) and the surface \(z=4-x^{2}-y^{2}, z \geq 0\), oriented by outward-pointing normals.

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THEOREM 1 Stokes' Theorem Let S be a surface as described earlier, and let F be a vector field whose components have continuous partial derivatives on an open region containing S. dr = ff curl(F). ds fF.dr = las The integral on the left is defined relative to the boundary orientation of aS. If S is a closed surface, then Iss curl(F)-dS=0