If $\displaystyle E, F, G$ are the coefficients of the first fundamental form in a parametrization $\displaystyle \mathbf{x}:U \subset \mathbb{R}^2 \mapsto S$, then $\displaystyle grad \ f$ on $\displaystyle \mathbf{x}(U)$ is given by:

$\displaystyle grad \ f =\frac{f_uG-f_vF}{EG-F^2}\mathbf{x}_u +\frac{f_vE-f_uF}{EG-F^2}\mathbf{x}_v$

Now what I have come to realize is that we can represent $\displaystyle grad \ f$ as $\displaystyle (\mathbf{x}_u, \mathbf{x}_v)

\begin{pmatrix}

E & F\\

F & G

\end{pmatrix}^{-1} \begin{pmatrix}

f_u\\

f_v

\end{pmatrix}$

Now we get $\displaystyle \begin{pmatrix}

E & F\\

F & G

\end{pmatrix}^{-1} $ from the second fundamental form.

where we have

$\displaystyle N_u = a_{11}\mathbf{x}_u+a_{21}\mathbf{x}_v $

$\displaystyle N_v = a_{12}\mathbf{x}_u+a_{22}\mathbf{x}_v $

Now is there a way that I can incorporate these elements to get it to work out? Can I for instance carry on with the following $\displaystyle II_p(a) = \langle D (grad \ f(v)),v\rangle$ where $\displaystyle v$ is a tangent vector on a surface?