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I am reading through an excerpt on Lee's Introduction to Smooth Manifolds which describes local coordinates using 'component functions'.

Here is the specific quote from text:

Given a chart $(U,\phi)$ we call the set $U$ a coordinate domain of each of its points. The map $\phi$ is called a (local) coordinate map, and the component functions $(x^1, x^2, ... x^n)$ of $\phi$, defined by $\phi(p) = (x^1(p), x^2(p),...x^n(p))$ are called local coordinates on $U$.

I know that in this case, $\phi:U \to U'$ is a mapping of $U$ to $U'$, where $U'$ is a subset of $\mathbb R^n$.

My question is, what exactly are the component functions? Are they literal powers of some $x$ that is a member of $U$, or am I misinterpreting the notation? What are the component functions representing?

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    $\begingroup$ They are just $n$-different smooth real valued functions on $U$. $\endgroup$
    – Sumanta
    Sep 22, 2020 at 8:25
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    $\begingroup$ For every point $p$ of a manifold we can find an open neighborhood $U$ of $p$ and and an open subset $U'$ of $\Bbb R^n$ with a homeomorphism $\varphi:U\to U'$ such that compatibility etc. conditions hold, and we write $x^k:=\pi_k\circ \varphi:U\to \Bbb R$ where $\pi_k:\Bbb R^n\to \Bbb R$ is the projection on $k$-th component for each $k=1,...,n$. $\endgroup$
    – Sumanta
    Sep 22, 2020 at 8:31
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    $\begingroup$ There is no harm if you $x_k$ instead of $x^k$, alternatively you may use $x^{(k)}$. $\endgroup$
    – Sumanta
    Sep 22, 2020 at 8:38

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