Find an closed 1-form on $\mathbb{R}^2 \backslash (0,0)$ that is not exact.

Question

I need help with the following problem. I am not sure how start and I would be very appreciative if someone could help me with this (I believe easy?) example.

Thanks in advance for any help!

You can ask google, and it will take you there directly. – Julien Apr 14 '13 at 18:30 — Julien, Apr 14 '13 at 18:30

score 1 · Accepted Answer · answered Apr 14 '13 at 18:32

You want to find a closed form that is not exact (i.e. that $\omega \neq dF$ for some zero form), so you will be able to see it is sufficient to find a form so, when you integrate around a circle say, you get a nonzero number. Persuing this, by poincare's lemma, if must not be continuous at a point the circle bounds.

This is the beginning of the very exciting de Rham cohomology.

Find an closed 1-form on $\mathbb{R}^2 \backslash (0,0)$ that is not exact.

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