proof of closed differential forms on a simple connected domain

lemma 1.

Let γ0 and γ1 be two regular homotopic curves in D with the same end-points. Let σ:[0,1]×[0,1]D be the homotopy between γ0 and γ1 i.e.


Notice that we may (and shall) suppose that σ is regular too. In fact σ([0,1]×[0,1]) is a compact subset of D. Being D open this compact set has positive distance from the boundary D. So we could regularize σ by mollification leaving its image in D.

Let ω(x,y)=a(x,y)dx+b(x,y)dy be our closed differential form and let σ(s,t)=(x(s,t),y(s,t)). Define


we only have to prove that F(1)=F(0).

We have


Notice now that being ay=bx we have




Notice, however, that σ(s,0) and σ(s,1) are constant hence xs=0 and ys=0 for t=0,1. So F(s)=0 for all s and F(1)=F(0). ∎

Lemma 2.

Let us fix a point (x0,y0)D and define a function F:D by letting F(x,y) be the integral of ω on any curve joining (x0,y0) with (x,y). The hypothesis assures that F is well defined. Let ω=a(x,y)dx+b(x,y)dy. We only have to prove that F/x=a and F/y=b.

Let (x,y)D and suppose that h is so small that for all t[0,h] also (x+t,y)D. Consider the increment F(x+h,y)-F(x,y). From the definition of F we know that F(x+h,y) is equal to the integral of ω on a curve which starts from (x0,y0) goes to (x,y) and then goes to (x+h,y) along the straight segment (x+t,y) with t[0,h]. So we understand that


For the integral mean value theorem we know that the last integral is equal to ha(x+ξ,y) for some ξ[0,h] and hence letting h0 we have


that is F(x,y)/x=a(x,y). With a similar argument (exchange x with y) we prove that also F/y=b(x,y). ∎


Just notice that if D is simply connected, then any two curves in D 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
Canonical name ProofOfClosedDifferentialFormsOnASimpleConnectedDomain
Date of creation 2013-03-22 13:32:49
Last modified on 2013-03-22 13:32:49
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 9
Author paolini (1187)
Entry type Proof
Classification msc 53-00
Related topic SubstitutionNotation