Question on Divergence Theorem Evaluate $\iint F.ds$  
where $F= \frac{59}{3} x^3 \hat i + \frac{59}{3} y^3 \hat j + \frac{59}{3} z^3 \hat k$, and $S$ is the surface: 
$S= \{(x,y,z) | x^2 + y^2 +z^2 =9\}$ 
Please explain in detail how to get the answer.

Simple application of the Divergence theorem:
$$\iint F\cdot ds = \iiint \nabla \cdot  F dV$$
$$=\iiint \nabla \cdot (\frac{59}{3} x^3 \hat i + \frac{59}{3} y^3 \hat j + \frac{59}{3} z^3 \hat k) dV$$
$$=\iiint (59 x^2 \hat i + 59 y^2 \hat j + 59 z^2 \hat k) dV$$
$$=\iiint 59(x^2+y^2+z^2) dV$$
$$=59 \cdot \iiint r^2 \cdot r^2 \sin \theta \, dr \, d\theta \, d\phi$$
$$=59 \cdot \int_{0}^{3} r^4 \, dr \int_{0}^{\pi} \sin \theta \, d\theta \int_{0}^{2\pi} d\phi$$
$$=59 \cdot \frac{3^5}{5} \cdot 2 \cdot 2\pi $$
