Suppose $S\subset \mathbb{R}^3$ is a regular and compact surface. Let
- $f:S \to \mathbb{R}$ a function with $f(p)=\|p\|^2$ for every $p \in S$ and
- $X(u,v)$ be a parametrization of $S$.
My textbook says that the function $g=f \circ X(u,v)$ is:
$$g = \langle \, X(u,v)-X(u_0,v_0), N\circ X(u,v)\,\rangle $$
Where $N$ is the Gauss map.
I can't find how can $g$ be expressed this way, besides,
$$ g = f \circ X = \| X(u,v)\|^2=\langle \, X(u,v),X(u,v) \, \rangle $$
Maybe the textbook has a mistake?