Let (I=iint_{mathcal{S}} mathbf{F} cdot d mathbf{S}), where [ mathbf{F}(x, y, z)=leftlanglefrac{2 y z}{r^{2}},-frac{x z}{r^{2}},-frac{x y}{r^{2}}ightangle ] (left(r=sqrt{x^{2}+y^{2}+z^{2}}ight))

Question:

Let $I=\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}$, where
\[
\mathbf{F}(x, y, z)=\left\langle\frac{2 y z}{r^{2}},-\frac{x z}{r^{2}},-\frac{x y}{r^{2}}ightangle
\]
$\left(r=\sqrt{x^{2}+y^{2}+z^{2}}ight)$ and $\mathcal{S}$ is the boundary of a region $\mathcal{W}$.
(a) Check that $\mathbf{F}$ is divergence free.
(b) Show that $I=0$ if $\mathcal{S}$ is a sphere centered at the origin. Explain why the Divergence Theorem cannot, however, be used to prove this.

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