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.

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

Thanks in advance for any help!

Answer 1

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.