Use the Divergence Theorem to calculate \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given vector field and surface.

\(\mathbf{F}(x, y, z)=\left\langle x y, y z, x^{2} z+z^{2}ightangle, \mathcal{S}\) is the boundary of the unit sphere.

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 - Sw div(F) dv F-dS=