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

\(\mathbf{F}(x, y, z)=\left\langle x^{2}-z^{2}, e^{z^{2}}-\cos x, y^{3}ightangle, \mathcal{S}\) is the boundary of the region bounded by \(x+2 y+4 z=12\) and the coordinate planes in the first octant.

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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 JJ F. ds = !!!w div(F) dV 1