Let \(\mathbf{F}=\langle 0,-z, 1angle\). Let \(\mathcal{S}\) be the spherical cap \(x^{2}+y^{2}+z^{2} \leq 1\), where \(z \geq \frac{1}{2}\). Evaluate \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) directly as a surface integral. Then verify that \(\mathbf{F}=\operatorname{curl}(\mathbf{A})\), where \(\mathbf{A}=(0, x, x z)\) and evaluate the surface integral again using Stokes' Theorem.

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. kos T = SS F.dr= curl(F). dS The integral on the left is defined relative to the boundary orientation of aS. If S is a closed surface, then Ja curl(F). dS=0 1