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?