proof of closed differential forms on a simple connected domain
Let and be two regular homotopic curves in with the same end-points. Let be the homotopy between and i.e.
Notice that we may (and shall) suppose that is regular too. In fact is a compact subset of . Being open this compact set has positive distance from the boundary . So we could regularize by mollification leaving its image in .
Let be our closed differential form and let . Define
we only have to prove that .
Notice now that being we have
Notice, however, that and are constant hence and for . So for all and . ∎
Let and suppose that is so small that for all also . Consider the increment . From the definition of we know that is equal to the integral of on a curve which starts from goes to and then goes to along the straight segment with . So we understand that
Just notice that if is simply connected, then any two curves in with the same end points are homotopic. Hence we can apply Lemma 1 and then Lemma 2 to obtain the desired result. ∎
|Title||proof of closed differential forms on a simple connected domain|
|Date of creation||2013-03-22 13:32:49|
|Last modified on||2013-03-22 13:32:49|
|Last modified by||paolini (1187)|