# Computing the matrix representation of the second fundamental form of a cylinder

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?

• I think you may want $g(u,v) = (f(u)cos(v),f(u)sin(v),u)$ instead of $g(u,v) = (f(u,v)cos(v),f(u)sin(v),u)$; what do you think? Cool question, +1!!!. Cheers! – Robert Lewis Nov 15 '18 at 0:01
• You are right I will make an edit. – 3nondatur Nov 15 '18 at 0:03
• In any differential geometry text, you'll find a recipe for the second fundamental form in terms of second-order partial derivatives of $g$. You most definitely do not want to differentiate the Gauss map. – Ted Shifrin Nov 15 '18 at 0:14
• By the way I think you might want $\partial_1 = \partial g / \partial u$ and so forth, not $\partial_1 = \partial f / \partial u$ etc. – Robert Lewis Nov 15 '18 at 1:25
• Again you are right I made an edit. – 3nondatur Nov 15 '18 at 7:11