What is your question? From my understanding you're trying to express grad(f) using E, F, G and you did it. The result is correct so what are you really asking?
If are the coefficients of the first fundamental form in a parametrization , then on is given by:
Now what I have come to realize is that we can represent as
Now we get from the second fundamental form.
where we have
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 where is a tangent vector on a surface?
since grad(f) is a tangent vector it is expressible by the base vectors:
grad(f) = a Xu + b Xv, then we need to compute a and b by computing the inner products with Xu and Xv:
Xu . grad(f) = a Xu . Xu + b Xu . Xv
df( Xu) = a E + b F
fu = a E + b F.
Similarly:
fv = a F + b G
So we get a system of linear equations. Solve it we get
(a, b)' = ( E F; F G)^{-1} (fu, fv)'