I have been reading Rick Miranda's Book and i am confused as how he works with differential forms, because he says and i quote
"However, it is very convenient to simply use a single formula in one specified chart to define a meromorphic 1-form $w$, and let the burden fall to the reader to check that the formula transforms uniquely to give a meromorphic $1$-form on all of $X$.This way of of defining meromorphic $1$-forms is employed sistematically."
So what does he mean by this ? Is it that if i have a Riemann surface to define a form he will just gives an expression in one coordinate chart and we can see what it is in the others? So supose we have $X$ the smooth affine plane curve defined by $f(u,v)=0$ and he gives me the form $du$ then using the coordinate transition maps i can see that in the other chart it must be $g'(v)dv$ and hence we have our form defined ?
New edit: Alright so to determine the forms on the other coordinate charts we will the coordinate changing maps given by the charts but how do i know that i can extend this be defined on all the surface ? I might not be able to do this right ? Unless there is another chart that covers that point and there the local representation of the form is in fact holomorphic.
Thanks in advance.