Christoffel symbol transformation

Question

The Christoffel symbols transform like

$$\Gamma^{\prime a}_{bc} = \frac{\partial x^{\prime a}}{\partial x^d} \frac{\partial x^e}{\partial x^{\prime b}} \frac{\partial x^f}{\partial x^{\prime c}} \Gamma^{d}_{ef} - \frac{\partial x^d}{\partial x^{\prime b}} \frac{\partial x^e}{\partial x^{\prime c}} \frac{\partial^2 x^{\prime a}}{\partial x^d \partial x^e}$$

Now the second term can be written as

$$ \frac{\partial x^e}{\partial x^{\prime c}} \frac{\partial x^d}{\partial x^{\prime b}} \frac{\partial}{\partial x^d} \left(\frac{\partial x^{\prime a}}{ \partial x^e}\right) = \frac{\partial x^e}{\partial x^{\prime c}} \frac{\partial}{\partial x^{\prime b}} \left(\frac{\partial x^{\prime a}}{ \partial x^e}\right) \color{\red}{=} \frac{\partial x^e}{\partial x^{\prime c}} \frac{\partial}{\partial x^{e}} \left(\frac{\partial x^{\prime a}}{ \partial x^{\prime b}}\right) = \frac{\partial}{\partial x^{\prime c}} \delta^a_b = 0$$

I don't think this is correct. I guess that the problem is that derivatives of different coordinates don't commute, and so the mistake is in the step $2 \to 3$, where the equal sign is red, but that the previous are correct. Is this right?

Answer 1

Plese consider https://cosmo.nyu.edu/yacine/teaching/GR_2019/homeworks/solution2.pdf meeting very close your question.

1. In your first equation, on the left side, indexes b and c should be primed, I think.

2.From the link above it follows that first equal sign in your equation two is correct. The second, red is false as you mentioned, due to partial derivatives with respect to primed and unprimed coordinates does not commute.

Following given link one can very easy continue your equation two in full correct form.