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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?2011-05-18
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    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.2011-05-18
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    There is an analytic chart-based definition of meromorphic differential on page 25 of [these notes](http://www.dpmms.cam.ac.uk/%7Eagk22/RS-notes.pdf), but there are other approaches (especially on compact Riemann surfaces, which are algebraic curves) and other kinds of differentials (e.g. $C^\infty$).2011-05-18

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