Graduate New Adler book on GR: Why do these coefficients go to zero?

[peasg]

This is page 73 of the book. As you can see, the mixed derivatives with the affine connections vanish in the second term. Why does that happen? This is used to prove that the connections are not a tensor, and i figured you could also reason it out even without making those terms vanish.

OBS: The derivatives are avaliated at P, for the reason that this is obtained via a taylor series of the transformation coefficients.

 

[TSny]

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This is page 73 of the book. As you can see, the mixed derivatives with the affine connections vanish in the second term. Why does that happen?

The terms $\left( \dfrac{\partial^2 \bar x^j}{\partial x^l \partial x^i } \right)_P \Gamma^i_{pq} V^q dx^l dx^p$ have been dropped because they contain products $dx^l dx^p$. Therefore, these terms are second-order in the infinitesimal coordinate displacements. Only terms up to first order need to be kept.

 

[peasg]

The terms $\left( \dfrac{\partial^2 \bar x^j}{\partial x^l \partial x^i } \right)_P \Gamma^i_{pq} V^q dx^l dx^p$ have been dropped because they contain products $dx^l dx^p$. Therefore, these terms are second-order in the infinitesimal coordinate displacements. Only terms up to first order need to be kept.

Oh, that makes perfect sense. Thank you for your time!