Need urgent assistance(Gussian curvature and differentiable vector fields)

[SuperLouisa90]

Hi I have a very difficult problem where I know some of the dots but can't connect them :(

So therefore I hope that there is someone who can assist me (hopefully :))

1. The problem statement, all variables and given/known data

Let S be a surface with orientation N. Let V \subset S be an open set in S and let f: V\subset S \rightarrow \mathbb{R} be any nowhere zero differentiable function in V. Let v_1 and v_2 be two differentiable (tangent) vector fields in V such that at each point of V, v_1 and v_2 and that v_1 \land v_2 = N

Then prove that K = \frac{<d(fN)(V_1) \land d(fN)(V_2), fN>}{f^3}

p.s. there is also a question two but since this is so difficult I live that out for the time being hoping we can get to that later.

2. Relevant equations


3. The attempt at a solution

Here is what I know

Since S has the orientation N that according to do Carmo Geometry book means that it can be covered with a neighbourhood N.

From what I get is that

dN(v1) = cv1 + dv2 and dN(v2) = ev1 + fv2 but how do Carmo goes from that the above is a mystery to me. So therefore I hope there is someone who would help me understand what I am missing ?

Cheers
Louisa