Christoffel symbols

[Hastings]

I've done all but the last question- showing that LN-M^2 can be expressed in terms E,F, G and their derivatives. Here is what I tried:
Rearrange the first and last equation to make LN and nN the subjects. Then computed the dot product LnN.N=ln since N is a unit vector. In the dot product there is a X_{uu .Xvv term which I don't know how to express in terms of} E,F, G and their derivatives.

Thanks

 

[stapel]

For other viewers, the (tiny) text in the image appears to be as follows:

Show that the functions \(\displaystyle \Gamma^k_{ij}\) can be expressed in terms of \(\displaystyle E,\, F,\, G\) and their first order partial derivatives with respect to \(\displaystyle u\) and \(\displaystyle v\), and find the expression in the case that \(\displaystyle F\, =\, 0\).

Hence show that the Gauss curvature \(\displaystyle K\) of \(\displaystyle M\) is expressible in terms of \(\displaystyle E,\, F,\, G\) and their first and second order partial derivatives with respect to \(\displaystyle u\) and \(\displaystyle v\).

[You need not find the expression for \(\displaystyle K\). You may assume that \(\displaystyle K\) is given by \(\displaystyle K\, =\, \left(LN\, -\, M^2\right)/\left(EG\, -\, F^2\right)\), but you should explain briefly why \(\displaystyle EG\, -\, F^2\) is non-zero.]