Let $F(x,y,z)=5z^2+(\frac{5}{3}{y}^3+tan(z))j+(5x^{2}z+2y^2)k.$ Use the Divergence Theorem to evaluate ${\displaystyle \underset{S}{\overset{\phantom{}}{}}}F*dS$ where $S$ is the top half of the sphere $x^2+y^2+z^2=1$ oriented upwards.

${}_{S}F*dS=$

Hint: Note that S is not a closed surface. First compute integrals over $S_1$ and $S_2,$ where $S_1$ is the disk $x^2+y^{2}1,$ oriented downward, and $S_2=S_{1}S.$