# Visualizing Non-Zero Closed-Loop Line Integrals Via Sheets?

1. Dec 7, 2013

### bolbteppa

How do I visualize $\dfrac{xdy-ydx}{x^2+y^2}$?

In other words, if I visualize a differential forms in terms of sheets:

and am aware of the subtleties of this geometric interpretation as regards integrability (i.e. contact structures and the like):

then since we can interpret a line integral as counting the number of sheets you cross through:

we see we can interpret the notion of a closed loop line integral as not being zero in terms of this contact structure idea, i.e. as you do the closed line integral you do something like enter a new field of sheets, what exactly are you doing & how does this explain the non-zero line integral around a closed loop at the origin for the differential form I've given above? Thanks!