Let $M \subset \mathbb{E}^3$ be a surface, $r:U \to M$ be its local parametrization, $f: M \to N \subset \mathbb{E}^3$ a diffeomorphism of manifolds. This induces a parametrization of $s: U \to N$.

The parametrizations $r, s$ induce bases of the tangent spaces of $M$ and $N$ respectively with $$r_i = \frac{\partial r}{\partial u_i}, s_i =\frac{\partial s}{\partial u_i}$$

Suppose that the first fundamental forms on $M$ and $N$ in this bases have the matrix $[g_{ij}]$ and $[h_{ij}]$ respectively. Can we somehow express the coefficients $h_{ij}$ in terms of $g_{ij}$ and $f$? The same question for the second fundamental form.

This looks like some chain rule, but I have trouble writing down the relevant partial derivatives of $f$. I can't just write $\sum_i\frac{\partial f}{\partial r_i}\frac{\partial r_i}{\partial u_i}$.