Let $X$ be a Riemann surface.
On p.106 of "Algebraic Curves and Riemann Surfaces" Rick Miranda writes, when proving that it suffices to specify a holomorphic $1$-form on $X$ on any (not necessarily maximal) atlas:
Let $\psi$ be a chart of $X$ not in a given atlas $A$; our task is to define the holomorphic $1$-form with respect to $\psi$ or, equivalently, in terms of the local coordinate $w$ of $\psi$.
What is a "local coordinate of a chart" (i.e. what is meant by "local coordinate $w$ of $\psi$" in this context)? Is $w$ another symbol for $\psi$? Is $w$ a function? If it is, I am confused about Miranda's subsequent notation "$f(T(w))T'(w)$". This notation makes it seem as if $w$ was a point in $\mathbb{C}$.
In the terminology I know, local coordinates are the coordinate functions of a chart (i.e. the composition of a chart with a projection). Here, we are considering charts from open subset of $\mathbb{C}$ to open subsets of $\mathbb{C}$. Thus, local coordinates would simply be the same as charts.
Best Answer
It seems to me that Miranda's terminology may be a little confusing.
He defines a (complex) chart on $X$ to be a homeomorphism $\phi : U \to V$, where $U \subset X$ is an open set in $X$, and $V \subset \mathbb C$ is an open subset in $\mathbb C$.
Then he writes
My understanding is that the concept of local coordinate is not identical with the concept of chart:
A chart is a map $\phi : U \to V$, but a local coordinate is an attribute of a point $x \in U$ (namely its image $\phi(x) \in V \subset \mathbb C$.
The connection is obvious: Each chart on $X$ gives a local coordinate to all points of its domain $U$, and we can reconstruct the chart if we know the local coordinates of all points in $U$.