ChapterZero

1-forms and topology

I ended up not reading much algebraic topology on the way here (trying to wend your way through proofs on a bus, even ones that have already been worked out for you, turned out to be slow and unfruitful work), but I’ve started reading Fulton’s Algebraic Topology since I arrived.

One of the basic issues in algebraic topology seems to be the connection between integration (or more generally, any sort of analysis process) and the topology of a manifold. As an exemplar question: when can a given 1-form $\omega = p(x,y) dx + q(x,y) dy$ be written as the differential of a function , $\omega = df$ ?

I don’t know yet, and the suspense is killing me. In the meantime, here’s a problem: note that if $f = \tan^{-1}(y/x)$ , then in the right half plane $df = \frac{-y dx + x dy}{x^2 + y^2}$ . However, the one form $\omega_\nu = \frac{-y dx + x dy}{x^2 + y^2}$ cannot be written as the differential of a function if the domain of definition is $\R^2 \setminus (0,0)$ . What if the domain of definition is the set
$\{(re^t \cos(t), r e^t \sin(t)), 0<t<4\pi, \frac{1}{2} < r < 2\}$ ?

This entry was posted on Friday, July 27th, 2007 at 9:44 am and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

1-forms and topology

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