<s>[INST] Verify the Divergence Theorem The vector field $  \ \large \vec F=(xy,yz,zx) \ $ , on the closed cylinder $ \ x^2+y^2 \leq 1 , \ \ z=0 , \ \ z \leq 1$. 
Verify the Divergence Theorem. 
Answer:
$ (i) $ we have to calculate here $ \iint_S F \cdot n \ dS$ 
Let $ z=1-x^2-y^2 \ $
Then $ \vec F=(xy,y(1-x^2-y^2) , x(1-x^2-y^2)) \ $
Now , 
$$ F_x=(y,-2xy, 1-3x^2) , \ \ F_y=(x, 1-x^2-3y^2, -2xy) $$
Then, 
$$ F_x \times F_y=(13x^2y^2+3x^4+2x^2-3y^2, \ 2xy^2+x-3x^3, \  x^2y+y-3y^3) $$
Then, 
$$ \iint_S F \cdot n \, dS= \int_{-1}^1 \int_{-1}^1 \vec F \cdot (F_x \times F_y) \, dx\,dy$$
But this becomes complicated .
I am confused . 
Am I right so far . 
Help me out
 [/INST] You should have started with choosing the paramaterization of your surface. We have three parts. We have the upper part the side part and the bottom part. I only do the side part. A paramterization for the side part is:
\begin{align}
\phi(t,\theta) =
\begin{pmatrix}
\cos \theta\\
\sin\theta\\
t
\end{pmatrix}
\end{align}
$t\in [0,1]$ and $\theta \in[0,2\pi]$. Then the normal vector is:
\begin{align}
\partial_\theta\phi(t,\theta) \times \partial_t\phi(t,\theta) = 
\begin{pmatrix}
-\sin\theta \\
\cos\theta\\
0\\
\end{pmatrix} \times
\begin{pmatrix}
0\\
0\\
1\\
\end{pmatrix}
=
\begin{pmatrix}
\cos\theta \\
\sin\theta \\
0
\end{pmatrix}
\end{align}
So:
\begin{align}
\iint_S \textbf{F}\cdot \textbf{n} dS &= \int^{2\pi}_0 \int^1_0 \textbf{F}(\phi(t,\theta))\cdot( \partial_\theta\phi(t,\theta) \times \partial_t\phi(t,\theta) ) dt d\theta\\
&=  \int^{2\pi}_0 \int^1_0 \cos^2\theta \sin\theta + t\sin^2\theta dtd\theta\\
&= \frac{\pi}{2}
\end{align}
For the last integral you need to use the trig identities as usual. As I already said I'll leave the top part and the bottom part for you. 
The divergence:
\begin{align}
\iiint_K \text{div} \textbf{F} dV 
\end{align}
Where $K=\{ (x,y,z) | x^2+y^2\leq 1 , 0\leq z\leq 1\} $.  Using polar coordinates:
\begin{align}
\iiint_K \text{div}\textbf{F} dV &=\iiint_K y+z+x dV\\
&=\int^1_0\int^1_0 \int^{2\pi}_0 (r\cos\theta+r\sin\theta + z)r d\theta dr dz\\
&= \frac{\pi}{2}
\end{align}
I assume integrating that is not a problem. Now do the bottom part and the top part of the surface integral to verify the Divergence Theorem. Let me know if there is something unclear.
</s>