Solution
= Solution
Write $h_x=\partial_xh$ and $h_y=\partial_yh$. Since
$$
\phi_x=(1,0,h_x),\qquad \phi_y=(0,1,h_y),
$$
the first fundamental form in the coordinate <basis> is
$$
\boxed{
I^S=
\begin{pmatrix}
1+h_x^2&h_xh_y\\
h_xh_y&1+h_y^2
\end{pmatrix}.}
$$