What does it mean by standard coordinates on $R^n$

Question

Here what does "standard coordinates" mean? Is it just the identity map of $R^n$? Or is it an arbitrary element of the standard smooth structure on $R^n$? The text is somewhat confusing. Could anyone please clarify it?

Answer 1

By "standard coordinates" on $\mathbb R^n$ it must be meant that points are identified by $(x_1,x_2,\ldots,x_n)$.

This corresponds to representing points in terms of the standard (ordered) basis for $\mathbb R^n$ as a vector space, $\mathbf e_1,\ldots,\mathbf e_n$, where $\mathbf e_k$ is the vector whose $k$th coordinate is one and the rest are zero.

Yes, it is effectively a complicated way of describing the identity map on $\mathbb R^n$, but it applies here to the treatment of $\mathbb R^n$ as an $n$-dimensional manifold.

The definition of an $n$-manifold requires an atlas of charts. In this case only a single chart will be needed, and for the sake of simplicity the identity map is used.