Fundamental forms of a graph surface
= Fundamental forms of a graph surface
For $X(x,y)=(x,y,h(x,y))$ and $W=\sqrt{1+h_x^2+h_y^2}$,
$$
I=
\begin{pmatrix}1+h_x^2&h_xh_y\\h_xh_y&1+h_y^2\end{pmatrix},
\qquad
II=\frac1W
\begin{pmatrix}h_{xx}&h_{xy}\\h_{xy}&h_{yy}\end{pmatrix}.
$$
The upward unit normal is $(-h_x,-h_y,1)/W$.