Christoffel Symbols and the change of transformation law. 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!
 A: 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 $\frac{\partial}{\partial x^m}$.
A: The second line is wrong. 
At first act with the transformation of the covariant derivative $Y^{j'}_{,p}$. Then you will get 
$$\frac{\partial x'^p}{\partial x^i}(Y^{j'})=\frac{\partial x'^p}{\partial x^i}(\frac{\partial x^{j'}}{\partial x^q} Y^p_q).$$
So, you will get a term that will cancel the last term in last line in your solution.
A: An a affine connection on a variety   $M$ is a map:
$$\nabla: \chi(M)\times \chi(M)\to \chi(M): (X, Y)\mapsto \nabla_XY$$
such that
$\nabla_{fX+gY}Z=f\nabla_XZ+g\nabla_YZ$.
$\nabla_{X}(Y+Z)=\nabla_XY+\nabla_XZ$.
$\nabla_{X}fY=X(f)\cdot Z+f\nabla_XZ$.
On a local chart with coordinate $x^1,\ldots x^m$  the  Christoffel symbols $\Gamma^k_{i,j}$ are defined by:
$$\Gamma^k_{i,j}\cdot e_k=\nabla_{e_i}e_j$$
where repeated indices are under summation and where  $e_i:=\frac{\partial}{\partial x^i}$.
Let's consider another coordinate system  $x'^1,\ldots x'^m$ on the some open set,  then:
$\Gamma'^t_{r,s}\cdot e'_t=\nabla_{e'_r}e'_s$ where $e'_s:=\frac{\partial}{\partial x'^s}$, then:
$\Gamma'^t_{r,s}\cdot e'_t=\nabla_{e'_r}e'_s=\nabla_{(\partial x^i/\partial x'^r e_i)}\ \partial x^j/\partial x'^s e_j=$
$ \partial x^i/\partial x'^r[\nabla_{e_i}\ (\partial x^j/\partial x'^s e_j)]=
 \partial x^i/\partial x'^r[e_i(\partial x^j/\partial x'^s )\cdot e_j + \partial x^j/\partial x'^s \cdot\nabla_{e_i}\ (e_j)]=$
$  \partial x^i/\partial x'^r[\partial^2 x^j/\partial x'^s\partial x'^u\cdot\partial x'^u/\partial x^i\cdot e_j + \partial x^j/\partial x'^s \cdot\Gamma^k_{i,j}\cdot e_k]=$
$  \partial x^i/\partial x'^r[\partial^2 x^j/\partial x'^s\partial x'^u\cdot\partial x'^u/\partial x^i\cdot \partial x'^t/\partial x^j\cdot e'_t + \partial x^j/\partial x'^s \cdot\Gamma^k_{i,j}\cdot \partial x'^t/\partial x^k\cdot e'_t]=$
$  [\partial^2 x^j/\partial x'^s\partial x'^u\cdot\partial x'^u/\partial x^i\cdot \partial x^i/\partial x'^r \cdot \partial x'^t/\partial x^j + \Gamma^k_{i,j}\cdot \partial x'^t/\partial x^k\cdot \partial x^j/\partial x'^s \cdot \partial x^i/\partial x'^r]\cdot e'_t$=
$  [\partial^2 x^j/\partial x'^s\partial x'^r \cdot \partial x'^t/\partial x^j+ \Gamma^k_{i,j}\cdot \partial x'^t/\partial x^k\cdot \partial x^j/\partial x'^s \cdot \partial x^i/\partial x'^r]\cdot e'_t$
then $$\Gamma'^t_{r,s}= \partial^2 x^u/\partial x'^s\partial x'^r \cdot \partial x'^t/\partial x^u+ \Gamma^k_{i,j}\cdot \partial x'^t/\partial x^k\cdot \partial x^j/\partial x'^s \cdot \partial x^i/\partial x'^r$$
$$\partial x^v/\partial x'^t\cdot \Gamma'^t_{r,s}= \partial^2 x^v/\partial x'^s\partial x'^r + \Gamma^v_{i,j}\cdot  \partial x^j/\partial x'^s \cdot \partial x^i/\partial x'^r$$
