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

\(\mathbf{F}(x, y, z)=\left\langle x+z^{2}, x z+y^{2}, z x-yightangle, \mathcal{S}\) is the surface that bounds the solid region with boundary given by the parabolic cylinder \(z=1-x^{2}\), and the planes \(z=0, y=0\), and \(z+y=5\).

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 J[ -JJJw div(F) dv F.dS= 1