I am reading "The Geometry of Physics: An Introduction" by Frankel, and am confused by the following construction in section 1.3a (perhaps available on Google Books using the Google query "from now on, we shall make no distinction between a vector"):
the $\alpha$-th coordinate curve is defined by $x^i(t)$=constant for $i\neq \alpha$ and $x^\alpha(t)=t$.
I do not understand this definition formally since the coordinates $x^\alpha$ are not functions of $t$!
Rather they are homeomorphisms like this
for some open sets $U$ and $V$ in the manifold $M$ and $(x_U,U)$ is a chart.. How can I make $x_U$ to be a function of a real parameter $t$?