I quote from W. Tu's introduction on manifolds:
«On a manifold $M$ of dimension $n$, let $(U, \varphi )$ be a chart and $f$ a $C^\infty$ function. As a function into $\mathbb{R}^n$ , $\varphi$ has $n$ components $x^1, \cdots, x^n$. This means that if $r^1 , \cdots , r^n$ are the standard coordinates on $\mathbb{R}^n$, then $x^i = r^i \circ φ$.»
My problem is that I do not understand why $x^i$ are not standard coordinates on $\mathbb{R}^n$. I suppose that the author has implicitly defined $x^i = \pi_i\circ \varphi$, where $\pi_i : \mathbb{R}^n \to \mathbb{R}$ is the function given by $\pi_i(x^1,\cdots,x^n) = x^i$. Could you explain this to me? Is the author confusing notation, i.e. $\pi_i = r^i$, or is he insisting that there exists some function $r^i \neq \pi_i$ for a $1 \leq i \leq n$?