Let \(\mathbf{F}(x, y, z)=\left\langle z^{2}, x+z, y^{2}ightangle\), and let \(\mathcal{S}\) be the upper half of the ellipsoid
\[
\frac{x^{2}}{4}+y^{2}+z^{2}=1
\]
oriented by outward-pointing normals. Use Stokes' Theorem to compute \(\iint_{\mathcal{S}} \operatorname{curl}(\mathbf{F}) \cdot d \mathbf{S}\).

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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. fF.dr = ffa dr = ff curl(F) - ds The integral on the left is defined relative to the boundary orientation of as. If S is a closed surface, then curl(F) curl(F)-dS=0