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

\(\mathbf{F}(x, y, z)=\left\langle x^{2} z, y x, x y zightangle, \mathcal{S}\) is the boundary of the tetrahedron given by \(x+y+z \leq 1,0 \leq x, 0 \leq y, 0 \leq z\).

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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 = [J/w div(F) dV 1