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

  1. Dec 7, 2013 #1
    How do I visualize [itex]\dfrac{xdy-ydx}{x^2+y^2}[/itex]?

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

    tITBK.png

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

    5dzXD.png

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

    1uqbK.png

    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!
     
  2. jcsd
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