I am trying to understand an example of a line bundle over a Riemann surface; as it is very terse and short, I have lots of trouble. It is written in block-quotes below, and I ask questions as I go.
The canonical bundle $K$ over a Riemann surface $M$ is the cotangent bundle, or the bundle of holomorphic $1$-forms. Suppose we have local coordinates $z$ and $w$ with $ w(z) = \phi_\beta \circ \phi_\alpha^{-1} (z) $ a function of $z$ on the overlap.
Question 1: What is meant by "with $w(z) = \phi_\beta \circ \phi_\alpha^{-1} (z)$ a function of $z$ on the overlap"? Why is it a function of $z$? I don't understand this equation at all.
The $1$-forms $dz$ and $dw$ give local trivializations of the canonical bundle, and on the overlap dw = w'dz.
Question 2: how are $dz$ and $dw$ are local trivializations? Isn't $dz$ at a point a map from the tangent space to $\mathbb{C}$? And how is this relation obtained?
Therefore the transition functions are $dw/dz$, where $w = \phi_\beta \circ \phi_\alpha^{-1}$.
Question 3: What does "dw/dz" even mean?