TITLE: Differential forms
QUESTION [14 upvotes]: I failed to understand the definition of holomorphic $1$-form on Riemann surfaces. Can one explain it here? I saw two definitions in the books of Miranda and Farkas-Kra.
Definition 1.: Suppose that $w_1=f(z)dz$ is a holomorphic $1$-form in the coordinate $z$, defined on an open set $V_1$. Also suppose that $w_2=g(w)dw$ be a holomorphic $1$-form in the coordinate $w$, defined on an open set $V_2$. Let $T$ be a holomorphic mapping from open set $V_2$ to $V_1$. We say that $w_1$ transforms to $w_2$ under $T$ if 
(1)    $g(w)=f(T(w))T'(w)$.
Let $X$ be a Riemann surface. A holomorphic $1$-form on $X$ is a collection of holomorphic $1$-forms $\{w_{\phi}\}$, one for each chart $\phi\colon U\rightarrow V $ in the co-ordinate of the target $V$, such that if two charts $\phi_i\colon U_i\rightarrow V_i$ (for $i=1,2$) have overlapping domains then the associated $1$-form $w_{\phi_1}$ transforms to $w_{\phi_2}$ under the change of coordinate mapping $T=\phi_1 \circ \phi_{2}^{-1}$.
Question 1: Can one explain the meaning of (1)? Can we say anything about it in terms of commutative diagram?

Definition 2. Let $M$ be a Remann surface. A $1$-form $w$ on $M$ is an (ordered) assignment of two continuous functions $f$ and $g$ to each local coordinate $z(=x+iy)$ on $M$ such that 
(2) $ fdx + gdy$
is invariant under coordinate changes; that is, if $\tilde{z}$ is another local coordinate on $M$ and the domain of $\tilde{z}$ intersects non-trivially with the domain of $z$, and if $w$ assigns the functions $\tilde{f},\tilde{g}$ to $\tilde{z}$, then 
(3) $ \tilde{f}(\tilde{z})=\frac{\partial x}{\partial \tilde{x}} f(z(\tilde{z})) + \frac{\partial y}{\partial \tilde{x}} g(z(\tilde{z})) $  
(4) $ \tilde{g}(\tilde{z})=\frac{\partial x}{\partial \tilde{y}} f(z(\tilde{z})) + \frac{\partial y}{\partial \tilde{y}} g(z(\tilde{z})) $  
Question 2 What is the meaning of these equations? How did they come (or how we imposed these conditions for the definition of holomorphic $1$-form?  Can we say anything about them in terms of commutative diagrams.

REPLY [8 votes]: The easiest way to understand differential 1-forms on a Riemann surface is to recall single variable calculus. For example, suppose $F$ is a function of one variable $x$. Then, its differential is
$$dF = F'(x) \, dx$$
and now suppose $x$ is itself a function of another variable $y$, say $x = T(y)$. Then,
$$dx = T'(y) \, dy$$
and so
$$dF = F'(T(y)) T'(y) \, dy$$
If we write $f = F'$ and $\omega = dF$ we get
$$\omega = f(x) \, dx = f(T(y)) T'(y) \, dy$$
which looks just like your first equation. This is the basic idea which motivates the abstract construction of differential 1-forms.
I don't think these equations are easily expressed in terms of commutative diagrams, as they are based on arbitrary choices of charts. An intrinsic definition of holomorphic differential 1-forms would be something along the lines of "holomorphic sections of the cotangent bundle". From a sheaf point of view, we're saying that the $\mathbb{C}$-space of holomorphic differential 1-forms defined on $U$, an open subset of the Riemann surface $S$, $U$ biholomorphic to $V$, an open subset of the complex plane $\mathbb{C}$, is isomorphic to the $\mathbb{C}$-space of holomorphic functions on $V$, and the equations in your post define the restriction maps of the sheaf.