I Manipulating Christoffel Symbols: Questions & Answers

[whatisreality]

I have a couple of questions about how Christoffel symbols work. Why can they just be moved inside the partial derivative, as shown just beneath the first blue box here: https://einsteinrelativelyeasy.com/index.php/general-relativity/61-the-riemann-curvature-tensor

And if you had the partial derivative:
$(\partial _{\rho} \Gamma^{\mu}_{\sigma\nu}) U^{\nu}$
Where $U^{\nu}$ is any contravariant vector, does this commute? Could I just move the $U^{\nu}$ from the right of the bracket to the left of the bracket?

 

[Orodruin]

I have a couple of questions about how Christoffel symbols work. Why can they just be moved inside the partial derivative, as shown just beneath the first blue box here: https://einsteinrelativelyeasy.com/index.php/general-relativity/61-the-riemann-curvature-tensor

The first blue box reads
$$
\Gamma_{\alpha\beta}^\gamma \vec e_\gamma = \frac{\partial \vec e_\beta}{\partial x^\alpha}.
$$
What do you mean by "just be moved inside the partial derivative"? There is no partial derivative on the left-hand side where the Christoffel symbol appears.

And if you had the partial derivative:
$(\partial _{\rho} \Gamma^{\mu}_{\sigma\nu}) U^{\nu}$
Where $U^{\nu}$ is any contravariant vector, does this commute? Could I just move the $U^{\nu}$ from the right of the bracket to the left of the bracket?

The partial derivative in this case acts only on the Christoffel symbol and $(\partial_\rho \Gamma^\mu_{\sigma \nu})$ should be treated as a number. You can freely move the $U^\nu$ around it.

 

[whatisreality]

The first blue box reads
$$
\Gamma_{\alpha\beta}^\gamma \vec e_\gamma = \frac{\partial \vec e_\beta}{\partial x^\alpha}.
$$
What do you mean by "just be moved inside the partial derivative"? There is no partial derivative on the left-hand side where the Christoffel symbol appears.
The partial derivative in this case acts only on the Christoffel symbol and $(\partial_\rho \Gamma^\mu_{\sigma \nu})$ should be treated as a number. You can freely move the $U^\nu$ around it.

But in the line directly beneath that box, it looks to me like the christoffel symbol got moved inside the partial derivative. That might not actually be what's happening, I'm not sure of that step.