The velocity vector field of a fluid (in meters per second) is
\[
\mathbf{F}(x, y, z)=\left\langle x^{2}+y^{2}, 0, z^{2}ightangle
\]
Let \(\mathcal{W}\) be the region between the hemisphere
\[
\mathcal{S}=\left\{(x, y, z): x^{2}+y^{2}+z^{2}=1, \quad x, y, z \geq 0ight\}
\]
and the disk \(\mathcal{D}=\left\{(x, y, 0): x^{2}+y^{2} \leq 1ight\}\) in the \(x y\)-plane. Recall that the flow rate of a fluid across a surface is equal to the flux of \(\mathbf{F}\) through the surface.
(a) Show that the flow rate across \(\mathcal{D}\) is zero.
(b) Use the Divergence Theorem to show that the flow rate across \(\mathcal{S}\), oriented with an outward-pointing normal, is equal to \(\iiint_{\mathcal{W}} \operatorname{div}(\mathbf{F}) d V\). Then compute this triple integral.