What is the difference between local coordinates and standard coodinates?

Question

I am reading "An introduction to Manifolds" by Loring Tu. There it is written that we denote the standard coordinates in $R^n$ by $r_1,r_2,....r_n$. If $(U,\phi:U\to R^n)$ is a chart of a manifold, we let $x^i=r^i\circ\phi$ to be the ith component of $\phi$ and write $\phi=(x^1,...x^n)$ and $(U,\phi)=(U,x^1,...x^n)$ . My question is why they've used $r_1,r_2,....r_n$ and $(x^1,...x^n)$ separately? And what does the ith component of $\phi$ mean? And what is the difference between local coordinates and standard coordinates? I am also attaching the screenshot where this is written.

Answer 1

Let's go through these questions one-by-one. Let's say that we're working in $\mathbb{R}^3,$ and we have a point $p=(p_1, p_2, p_3).$ Then, $r^i(p)=p_i.$ That is, $r^i$ just extracts the $i$th component of your point in $\mathbb{R}^3.$ Your chart takes a point on the manifold and returns a point in $\mathbb{R}^n,$ and what $x^i$ does is simply take the $i$th coordinate of this point. That is, if $p\in U\subset M,$ then $x^i(p)=r^i (\phi(p))$. Observe that $\varphi(p)\in\mathbb{R}^n,$ and then $r^i$ simply takes the $i$th component (since $\varphi(p)$ lives in $\mathbb{R}^n$, it is a vector with $n$ components). So, these functions give coordinates in a manner analogous to what we already understand in Euclidean space.

Now, let me explicitly answer your final question. The standard coordinates on $\mathbb{R}^n$ are the coordinate functions in $\mathbb{R}^n$. Local coordinates on $U$ are the coordinate functions defined by your chart in $\mathbb{R}^n$. That is, the components of your chart are translated into Euclidean coordinates.