I have seen it written that the change of co-ordinate form is given by the following:
$$ \tilde \Gamma^{i}_{jk} = {\partial \tilde x^i \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta \gamma}{\partial x^\beta \over \partial \tilde x^i}{\partial x^\gamma \over \partial \tilde x^k} + {\partial ^2 x^\alpha \over \partial \tilde x^j \partial \tilde x^k} \right ]$$
I am at a loose end as to how to prove this though. I have seen a proof(which I have included below) but in particular I do not understand the second step.
If someone could explain this second step, I would be very grateful!
Note : I would only like to see a proof involving co-ordinates. If you prove another way, please be sure to include some detail, as I will most likely be unfamiliar with it!
Thanks!
Best Answer
I'll go over the proof you included one line at at time: The first equality just uses the definition of the covariant derivative. The second equality uses the assumption that $\nabla_i Y^j$ is a tensor; this is the standard transformation law for any (1,1) tensor under coordinate transformations. The next equality is a similar step. Since $Y^l$ transforms as a tensor too, the step here is to re-express $Y^\prime$ in terms of $Y$. Then it's just expansion of the partial derivative $\frac{\partial}{\partial x^m}$.