Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).

\(\mathbf{F}(x, y, z)=\left\langle x^{3}, 0, z^{3}ightangle, \mathcal{S}\) is the boundary of the region in the first octant of space given by \(x^{2}+y^{2}+z^{2} \leq 4\), \(x \geq 0, y \geq 0, z \geq 0\).

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THEOREM 1 Divergence Theorem Let S be a closed surface that encloses a region W in R. Assume that S is piecewise smooth and is oriented by normal vectors pointing to the outside of W. If F is a vector field whose components have continuous partial derivatives in an open domain containing W, then 1 F. ds = [//w div(F) dv 1