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

Question

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!

Answer 1

First, it is important to note that those are oriented sheets: they have two sides, one painted red and another blue. When a curve crosses a sheet, the red-blue crossing counts as $1$ while the blue-red crossing counts as $-1$.

Picture a stack of vertical planes passing through the $z$-axis, like a book that opens 360 degrees wide. Here is an example. Odd-numbered pages are red, even-numbered are blue. A loop around the spine of the book crosses all pages in the same direction (either red-blue or blue-red). There is no cancellation, hence the integral is nonzero.