I try to calculate the second fundamental form of cylindrical graphs. Let $$ B= \{(x_1,x_2,x_3) \in \mathbb R^3 : x_1^2 + x^2_2 =1 \} $$ is the unit cylinder. Define the height function $$ u:B\rightarrow \mathbb R^+ $$ The cylinderical graph is $$ M= \{ (y_1,y_2,y_3) \in \mathbb R^3 : y_1 = u x_1,~ y_2 = ux_2 ,~ y_3 =x_3 ,~ \forall (x_1,x_2,x_3) \in B \} $$
Let $x_1 = \sin \theta ,~ x_2 =\cos \theta ,~~\theta\in[0,\pi)$. Then we have $$ y_\theta= ( u_\theta\cos\theta-u\sin\theta, u_\theta\sin\theta+u\cos\theta, 0 ) \\ y_{x_3}= ( u_{x_3}\cos\theta, u_{x_3}\sin\theta,1 ) $$ As general way, I should calculate the normal vector $\nu$ by $y_\theta$ and $y_{x_3}$. And then calculate $y_{\theta\theta}$ , $y_{\theta x_3}$ and $y_{x_3x_3}$. But this a cylinderical graph, I think $$ y_{\theta\theta} \cdot \nu ~~~~~ y_{\theta x_3}\cdot \nu ~~~~~ y_{x_3x_3}\cdot \nu ~~~~~ $$ is not the second fundamental form of $M$, since $(\theta, x_3)$ is not a Eucildean space. So, how should I do ?