Christoffel symbols transformation law

[guitarphysics]

In Carroll's GR book (pg. 96), the transformation law for Christoffel symbols is derived from the requirement that the covariant derivative be tensorial. I think I understand that, and the derivation Carroll carries out, up until this step (I have a very simple question here, I believe- something stupid I'm not seeing):
$$\Gamma^{\nu'}_{\mu'\lambda'} \frac{\partial x^{\lambda'}}{\partial x^{\lambda}}V^{\lambda} + \frac{\partial x^{\mu}}{\partial x^{\mu'}}V^{\lambda}\frac{\partial}{\partial x^{\mu}}\frac{\partial x^{\nu'}}{\partial x^{\lambda}} = \frac{\partial x^{\mu}}{\partial x^{\mu'}}\frac{\partial x^{\nu'}}{\partial x^{\nu}} \Gamma^{\nu}_{\mu\lambda}V^{\lambda}$$

Since this must be true for any vector $V^{\lambda}$, that can be eliminated. We can then multiply by $\partial x^{\lambda}/\partial x^{\sigma'}$ on both sides, and relabel $\sigma' \to \lambda'$ to get:
$$\Gamma^{\nu'}_{\mu'\lambda'} = \frac{\partial x^{\mu}}{\partial x^{\mu'}}\frac{\partial x^{\lambda}}{\partial x^{\lambda'}} \frac{\partial x^{\nu'}}{\partial x^{\nu}} \Gamma^{\nu}_{\mu\lambda} + \frac{\partial x^{mu}}{\partial x^{\mu'}}\frac{\partial x^{\lambda}}{\partial x^{\lambda'}}\frac{\partial^2 x^{\nu'}}{\partial x^{\mu} \partial x^{\lambda}}$$

Now, my question is just this: why the hell is there a plus sign in the last equation, instead of a minus sign? If I follow Carroll's steps directly from the first equation, I get a minus sign! :(
Sorry if it's obvious- any help is appreciated!

 

[Mr-R]

I don't have his book on me but I looked up his notes online here https://preposterousuniverse.com/wp-content/uploads/grnotes-three.pdf (page 56)and the sign is negative.
But check again if you are writing in terms the primed/non-primed coordinates and which you initially started with. I am sure its a small algebraic/relabeling issue.

 

[George Jones]

Now, my question is just this: why the hell is there a plus sign in the last equation, instead of a minus sign? If I follow Carroll's steps directly from the first equation, I get a minus sign!

My copy of Carroll's book has a minus sign (in (3.10) on page 96).

 

[guitarphysics]

Which one is correct? For instance, if you look here https://en.wikipedia.org/wiki/Christoffel_symbols#Change_of_variable
they have it with a plus sign. What year is your book from? I think mine's from 2004.

 

[George Jones]

As Mr-R noted, be careful with the indices!

I have to catch my bus now, but I will try to type in the details after I get home tonight.

 

[George Jones]

Here are the quantitative details:

$$\begin{align}
\frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\frac{\partial x^{\lambda }}{\partial x^{\lambda ^{\prime }}}\frac{\partial ^{2}x^{\nu ^{\prime }}}{\partial x^{\mu }\partial x^{\lambda }} &= \frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\left( \frac{\partial x^{\lambda }}{\partial x^{\lambda ^{\prime }}}\frac{\partial }{\partial x^{\lambda }}\right) \frac{\partial x^{\nu ^{\prime }}}{\partial x^\mu} \\
&= \frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\left( \frac{\partial }{\partial x^{\lambda ^{\prime }}}\frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu }}\right) \\
&= \frac{\partial }{\partial x^{\lambda ^{\prime }}}\left( \frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu }}\right) -\frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu }}\left( \frac{\partial }{\partial x^{\lambda ^{\prime }}}\frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\right) \\
&= \frac{\partial }{\partial x^{\lambda ^{\prime }}}\left( \frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu ^{\prime }}}\right) -\frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu }}\left( \frac{\partial }{\partial x^{\lambda ^{\prime }}}\frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\right) \\
&= \frac{\partial }{\partial x^{\lambda ^{\prime }}}\left( \delta _{\mu ^{\prime }}^{\nu ^{\prime }}\right) -\frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu }}\left( \frac{\partial }{\partial x^{\lambda ^{\prime }}}\frac{\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}\right) \\
&= -\frac{\partial x^{\nu ^{\prime }}}{\partial x^{\mu }}\frac{\partial ^{2}x^{\mu }}{\partial x^{\lambda ^{\prime }}\partial x^{\mu ^{\prime }}} .
\end{align}$$

Either the expression at the beginning or the term at the end can be used. Not the differing signs (and indices!).

 

[guitarphysics]

Sorry for the late response, I had been procrastinating going over the indices :s
I'm still not really seeing how that affects the original post; the last term you end on is not present, I believe, in Carroll's derivation at all. (It probably is and I'm just not seeing it- in which case I apologize!)