TITLE: Why Differential Forms on Riemann surfaces?
QUESTION [5 upvotes]: I am working with Rick Miranda's "Algebraic Curves and Riemann Surfaces".
Right now I am in chapter four "Integration on Riemann Surfaces" and struggle with it a lot!:(
It starts with the definition of holomorphic 1-forms which is as follows:
Definition: A holomorphic 1-form on an open set $V\subset \mathbb C$ is an expression $\omega$ of the form
$$\omega=f(z)\text{dz}$$
where $f$ is a holomorphic function on $V$. We say that $\omega$ is a holomorphic 1-form in the coordinate z.
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The main problem for me is now that i can't really work with this expression. What is $\omega=f(z)\text{dz}$? Why should we define something like this/ what is the benefit from that? 
We already have a notion for the integral of a function on submanifolds. Why we should not apply this here?
Because of this open question I am not able to understand the definitions which come later for example the following two:
Definition: A $C^{\infty}$ 1-form on an open set $V\subset \mathbb C$ is an expression $\omega$ of the form
$$\omega=f(z,\overline{z})\text{dz}+g(z,\overline{z})\text{d}\overline{\text{z}}$$
where $f$ and $g$ are $C^{\infty}$ function on $V$. We say that $\omega$ is a $C^{\infty}$ 1-form in the coordinate $z$.
I think my problems with this definition are the same as above.
Definition: A $C^{\infty}$ 2-form on an open set $V\subset \mathbb C$ is an expression $\eta$ of the form 
$$\eta=f(z,\overline{z})\text{dz}\wedge\text{d}\overline{\text{z}}$$
where $f$ is a $C^{\infty}$ function on $V$. We say that $\eta$ is a $C^{\infty}$ 2-form in the coordinate $z$.
My additional problem/question here is: What is $\wedge?$. At this point its the first time that the author introduces this symbol.
I would be very very glad if someone can make this clear for me. I appreciate any kind of help.

REPLY [2 votes]: My understanding is when we do integration, we are not integrating functions but forms instead. What eats vectors from the tangent bundle of surface/manifold is things living in cotangent bundle. So it's $dz$ part that deals with vectors coming from $ \Omega$ and yield number out to reflect orientation and infinitesimal area.
As to the mysterious wedge product, what helps us understand it may be the simple version relation between volume of space spanned by vectors and derterminant of corresponding matrix. The antisymmetric property of derterminant may give a hint why we also need antisymmetic property for forms.