uses the Stokes theorem to calculate the surface integral, $I=\int_{S}CurlA.ds$, where $A(x,y,z)=(2y,3x,-z^{2})$, for all $x,y,z\in \mathbb{R}$ and $S=\left\{(x,y,z) \in \mathbb{R^{3}}: x^{2}+y^{2}+z^2=9, 0\leq z\leq 1\right \}$.
Assume a parameterization of $S$ in which the normal field moves away from the $z$-axis.
I would like them to help me indicating how to do the parameterization.
You don't need the parameterization of $S$. You just need to know that the normal points aways from the $z$-axis. You will be computing the circulation about the boundary, and knowing the direction of the normal tells tells what the orientation of the boundary curve (or curves in this case).
Paul's Online Math Notes explains about the orientation.
It's the boundary curves that you need to parameterize, but they're just circles in this problem, so it's easy. Be careful though. One circle will be clockwise and one counterclockwise.