Let $S={(x,y,z)\in \mathbb{R}^3 : z=x^2-y^2,x^2+y^2=\leq 1}$ oriented upwards. Find the flux of $F(x,y,z)=(x,y,1)$ through $S$.
With the parametrization $x(r,\theta)=(r\cos\theta,r\sin\theta,r^2\cos2\theta), 0\leq r\leq1,0\leq \theta\leq 2\pi$ we get
$$\iint_S F dS= \int_0^1 \int_0^{2\pi}-2r^3\cos2\pi + r \ \ d\theta dr= \pi$$
But then
$$\iiint_R \text{div }F \ dV=2 \int_0^1 \int_0^{2\pi} 2r^3 \cos 2\pi \ d\theta dr=0$$
Where did I go wrong???
When you apply the divergence theorem, you are computing the flux across the total boundary of the region $R$. Since you didn't write down the triple integral, it's hard to read your mind. (So you should be including lots more details if you expect us to find your errors!) What is the region $R$? Does it not include a base and cylindrical sides, along with the surface $S$ on the top?