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):(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \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} [/tex]

Since this must be true for any vector [itex] V^{\lambda} [/itex], that can be eliminated. We can then multiply by [itex] \partial x^{\lambda}/\partial x^{\sigma'} [/itex] on both sides, and relabel [itex] \sigma' \to \lambda' [/itex] to get:

[tex] \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}} [/tex]

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!

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# I Christoffel symbols transformation law

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