|
|
|
|
closed differential forms on a simply connected domain
|
(Theorem)
|
|
|
Let $D\subset \mathbb R^2$ be an open set and let $\omega$ be a differential form defined on $D$
The proof of this result is a consequence of the following useful lemmas.
Lemma 1 Let $\omega$ be a closed differential form and suppose that $\gamma_0$ and $\gamma_1$ are two regular homotopic curves in $D$ (with the same end points). Then $$ \int_{\gamma_0} \omega = \int_{\gamma_1}\omega. $$
Lemma 2 Let $\omega$ be a continuous differential form. If given any two curves $\gamma_0$ $\gamma_1$ in $D$ with the same end-points, it holds $$ \int_{\gamma_0} \omega = \int_{\gamma_1} \omega, $$ then $\omega$ is exact.
See the Poincaré Lemma for a generalization of this result on $n$ dimensional manifolds.
|
"closed differential forms on a simply connected domain" is owned by paolini.
|
|
(view preamble | get metadata)
Cross-references: manifolds, Poincaré lemma, continuous, end points, curves, homotopic, regular, consequence, proof, exact differential form, closed differential form, simply connected, differential form, open set
There is 1 reference to this entry.
This is version 11 of closed differential forms on a simply connected domain, born on 2003-04-04, modified 2004-06-17.
Object id is 4146, canonical name is ClosedDifferentialFormsOnASimpleConnectedDomain.
Accessed 2852 times total.
Classification:
| AMS MSC: | 53-00 (Differential geometry :: General reference works ) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
