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

\(\mathbf{F}(x, y, z)=\langle x+y, z, z-xangle, \mathcal{S}\) is the boundary of the region between the paraboloid \(z=9-x^{2}-y^{2}\) and the \(x y\)-plane.

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