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I am studying general relativity through the wonderful book: "General Relativity: An Introduction for Physicists" by M.P. Hobson (Cambridge University Press) (2006). My question is about Riemannian manifolds and local cartesian coordinates (Chapter 02 - Section 2.11 - Page 43). Section starts explaining that for a general Riemannian manifold (considering only strictly Riemannian manifolds), it is not possible to perform a coordinate transformation from an arbitrary coordinate system to a desired coordinate system that will take the line element into a Euclidean form. Section goes on to explain that, however, it is possible to make a coordinate transformation such that in the neighbourhood of some especified point P the line element takes the Euclidean form. The next step is to demonstrate that fact by applying the expansion in Taylor series about the point P. The general transformation rule for the metric functions is called to perform the Taylor expansion. My question is this demonstration. In particular I am not able to understand the expression provided for expansion in Taylor series, especially terms that have derivatives of higher orders. I tried to justify this fact by considering an open ball centered on the point P, where the higher order derivatives arises from points in the neighbourhood of the point P. To be honestly I am not convinced that I am on the right track. Could someone explain me this demonstration, in particular the expansion in Taylor series. I am having doubts on writing Taylor expansion about the point P.

I am really grateful for any response.

PS.: Unfortunately, I still can not write equations using Latex on our forum, which would facilitate the understanding of my doubt.