I am a first year undergraduate teaching myself General Relativity from the book by Bernard Schutz. In one of the problems he asks to derive the transformation law for the Christoffel symbols from the definition:
\begin{equation} \Gamma^{\mu}_{\alpha\beta} \vec{e}_{\mu} = \frac{\partial \vec{e}_{\alpha}}{\partial x^\beta}.\tag{1} \end{equation} After a lot algebra and using the transformation laws for the 'known' quantities, I arrived at: \begin{equation} \Gamma^{\mu '}_{\alpha '\beta '} = \frac{\partial x^{\mu'}}{\partial x^{\gamma}}\frac{\partial^2x^\gamma}{\partial x^{\beta '}\partial x^{\alpha '}}+\frac{\partial x^{\mu'}}{\partial x^\gamma}\frac{\partial x^{\sigma}}{\partial x^{\alpha '}}\frac{\partial x^p}{\partial x^{\beta '}}\Gamma^{\gamma}_{\sigma p}.\tag{2} \end{equation} Now Wikipedia says the order of partial derivatives is swapped in the first term. But the expression that I am getting through first principles is the one above. However Schutz mentions towards the end of the chapter that for a non-coordinate basis the definition of the Christoffel symbol changes and hence the one that was used to derive the above is only valid for coordinate basis in which the partial derivatives can be swapped. So by that assumption, since the basis for this derivation is coordinate, I can swap the order of the partial derivatives above and obtain the expression: \begin{equation} \Gamma^{\mu '}_{\alpha '\beta '} = \frac{\partial x^{\mu'}}{\partial x^{\gamma}}\frac{\partial^2x^\gamma}{\partial x^{\alpha '}\partial x^{\beta '}}+\frac{\partial x^{\mu'}}{\partial x^\gamma}\frac{\partial x^{\sigma}}{\partial x^{\alpha '}}\frac{\partial x^p}{\partial x^{\beta '}}\Gamma^{\gamma}_{\sigma p}.\tag{3} \end{equation} which is the one in standard texts and Wikipedia. Am I correct in reasoning so? I apologize if I sound naive as it is my first time learning the material.