Use the divergence theorem to evaluate

$$S

F $$ dS$,$

where

F$($x$,$ y$,$ z$)=$ z$2$x$$i $+$

y$33$

$+$ tan$1($z$)$

$$j $+($x$2$z $+$ y$2)$k

and S is the top half of the sphere

x$2+$ y$2+$ z$2=4.$

$[$Hint: Note that S is not a closed surface. First compute integrals over S$1$ and S$2,$ where S$1$ is the disk

x$2+$ y$24,$

oriented downward, and

S$2=$ S $$ S$1.]$