Question $5$

$($a$)$ The Divergence Theorem states that the flux of a vector field $F$ across a closed surface $S$

is equal to the triple integral of the

of $F$ over the region $D$ enclosed by $S.$

$($b$)$ Consider a region $D$ enclosed by the cylinder $x^2+y^2=9$ and the planes $z=0$ and $z=4.$

Express the region $D$ using

coordinates.

$($c$)$ Consider the vector field $F(x,y,z)=x^{2}i+y^{2}j+z^{2}k.$ Let $S$ be the surface of the region $D$

enclosed by the cylinder $x^2+y^2=4$ and the planes $z=0$ and $z=3.$ Use the Divergence

Theorem to calculate the flux of $F$ across the surface $S.$

$($d$)$ Describe two geometric applications of Divergence Theorem.