# Differential Form on Riemann Surface

Can one explain what is differential form on a Riemann Surface? What is motivation to define it?

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Do you know what a differential form on a smooth manifold is? –  Qiaochu Yuan May 18 '11 at 6:09
For one, differential forms provide a coordinate-invariant description of things to integrate over a Riemann surface. Unless you want to die a slow, painful death in notation bookkeeping, differential forms are your friend. –  Gunnar Magnusson May 18 '11 at 7:04
There is an analytic chart-based definition of meromorphic differential on page 25 of these notes, but there are other approaches (especially on compact Riemann surfaces, which are algebraic curves) and other kinds of differentials (e.g. $C^\infty$). –  Zhen Lin May 18 '11 at 8:41