How to find the matrix for $dN_p$, the differential of the Gauss map?

Question

Suppose that $x:U\rightarrow \mathbb{R}^3$ is a chart for a regular surface $S$. Using the notation (from Shifrin P.39, 46) that $N_p$ is the Gauss map at point $p$, whereas the matrix with $E$,$F$,$G$ and $l$,$m$,$n$ are the first and the second fundamental form respectively, how can I prove that the matrix for $dN_p$, with respect to the basis $\{\partial_ux,\partial_vx\}$, is given by $-\begin{bmatrix} E &F\\F&G \end{bmatrix} ^{-1}$$\begin{bmatrix} l &m\\m&n \end{bmatrix} $?

Answer 1

$$(dN\ x_u,x_u)= (N,x_u)_u - (N,X_{uu})= -(N,x_{uu})=-l $$

First equality is the differential of $(N,x_u)$

And second equality is followed from $(N,x_u)=0$ and third is definition of $l$

If $dN\ x_u=N_{11}x_u + N_{21}x_v$, then $$ N_{11}E+N_{21} F=-l $$ where $N_{ij}$ is entry of $dN$. By considering other case we can prove it.