Let $\Gamma:=\{(x,f(x)):x\in \mathbb{R}^n\}$ where $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is $C^\infty$. I am interested in the curvature of $\Gamma$ given by each points at $x\in \mathbb{R}^n$. I know the mean curvature $nH=\sum_{i=1}^n\kappa_i$ (where $\kappa_i$ are principal curvatures, eigenvalues of the second fundamental form) is given by the next formula: $$nH=\frac{1}{\sqrt{1+|\nabla f|^2}}\nabla\cdot\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^2}}\right).$$ I would like to have a formula for the second fundamental form, and my ansatz is $$II=\frac{1}{\sqrt{1+|\nabla f|^2}}\nabla\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^2}}\right)$$ $$=\frac{\nabla^2f}{1+|\nabla f|^2}+\frac{\nabla f\otimes \nabla f}{(1+|\nabla f|^2)^2},$$ so after a linear transformation $II$ is exactly the Hessian of $f$.
Question: Is this guess correct? I did some search but could not find a reference. So it will be appreciated if I can have any reference, too.
I am sorry about my ignorance on the topic, I didn't have a change to learn Riemannian geometry yet.