first fundamental form: an example with $r_x=(1,0,f_x)$

Question

I cannot find the procedure on how, in Example I in the second snippet below, $$r_x=(1,0,f_x)$$ is found.

Answer 1

That is just the partial derivative $\frac{\partial}{\partial x}$ of the map

$$(x,y) \mapsto (x,y, f(x,y)).$$

Answer 2

$r$, the position vector, is always $(x,y,z)$, that is, $r=(x,y,z)$.

On this surface $z=f(x,y)$. So we can parametrize the surface by $x$ and $y$ coordinates.

Hence, on the surface, $r=(x,y,f(x,y))$.

So, $r_x=(1,0,f_x)$.