Abstract
This paper is to extend the Poincar’e Lemma for differential forms in a bounded, convex domain [1] in Rn to a more general domain that, we call, is deformable to every point in itself. Then we extend the homotopy operator T in [1] to the domain defromed to every point of itself.
Highlights
In [2], we have the Converse of the Poincar’e Lemma: Lemma 1.1
This paper is to extend the Poincar’e Lemma for differential forms in a bounded, convex domain [1] in Rn to a more general domain that, we call, is deformable to every point in itself
We extend the homotopy operator T in [1] to the domain defromed to every point of itself
Summary
In [2], we have the Converse of the Poincar’e Lemma: Lemma 1.1. Let U be a domain in Rn which can be deformed to a point P. Let ω be a (p+1)-form on U such that d 0. There is a p-form in U such that d. In [1] we have Lemma 1.2. Let D be a bounded, convex domain in Rn. To each y D there corresponds a linear operator. We extend the results of both of them. First we extend the bounded, convex domain D to the domain that deformed to every interior point. We gain that the closed form is the exact form, but every form can be decomposited to two parts where one of them is an exact form and another is a form related to the exterior differential of the form
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