Consider the generalised cylinder in $\mathbb{R}^3$ given by the parametrisation
$g(u,v) = (f(u)cos(v),f(u)sin(v),u)$
for $u \in (a,b) \text{ and } v \in (0,\pi).$ Find the matrix representation of the second fundamental form $II_x$ for any $x$ on the surface.
I know that the second fundamental form is given by $I(W_x X, Y)$, where $W$ means the Weingarten map. To compute this I need to get the Gauss map first, since the Weingarten map is it's derivative. To compute the Gauss map I suppose I should start like this:
$p\longmapsto \dfrac{\partial_1\times\partial_2}{||\partial_1\times\partial_2||}$, where $\partial_1=\frac{\partial g}{\partial u}$ and $\partial_2=\frac{\partial g }{\partial v}$.
Unfortunately I do not get any further, as I do not know how to concretely go forward. Could you help me?