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I've been trying to get some intuition for differential forms. I know the formal theory and I know how useful they are. But then I came across the following paper: https://dl.dropboxusercontent.com/u/828035/Mathematics/forms.pdf

It describes an intuition for forms that is very closely related to how forms are described in MTW. It's really nice.

The only problem is that I have no clue how to make this rigorous. So does anybody know some theorems or results that connect the intuitive picture as described in the pdf to the formal theory of forms. In particular, given a ##1##-form (or an ##n##-form more generally), how does one draw the associated picture in the pdf?

Thanks a lot!