# Component Functions of a Coordinate Map

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?

• They are just $n$-different smooth real valued functions on $U$. Sep 22, 2020 at 8:25
• 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 $k=1,...,n$. Sep 22, 2020 at 8:31
• There is no harm if you $x_k$ instead of $x^k$, alternatively you may use $x^{(k)}$. Sep 22, 2020 at 8:38