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In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose differential is zero (dα = 0), and an exact form is a differential form that is the differential of another differential form (α = dβ for some differential form β, known as a primitive for α).
Since d2 = 0, to be exact is a sufficient condition to be closed. The main interest of this pair of definitions, thus, is that asking whether this is also a necessary condition is a way of detecting topological information, by differential conditions. It makes no real sense to ask whether a 0-form is exact, since d increases degree by 1.
When the difference of two closed forms is an exact form, they are said to be cohomologous to each other. That is, if ζ and η are closed forms, and one can find some β such that
then one says that ζ and η are cohomologous to each other. Exact forms are sometimes said to be cohomologous to zero. The set of all forms cohomologous to each other form an element of a de Rham cohomology class; the general study of such classes is known as cohomology.
The cases of differential forms in R2 and R3 were already well-known in the mathematical physics of the nineteenth century. In the plane, 0-forms are just functions, and 2-forms are functions times the basic area element dx∧dy, so that it is the 1-forms
that are of real interest. The formula for the exterior derivative d here is
where the subscripts denote partial derivatives. Therefore the condition for α to be closed is
In this case if h(x,y) is a function then
The implication from 'exact' to 'closed' is then a consequence of the symmetry of second derivatives, with respect to x and y.
Poincaré lemma
The Poincaré lemma states that if X is a contractible open subset of Rn, any smooth closed p-form α defined on X is exact, for any integer p > 0 (this has content only when p ≤ n).
Contractability means that there is a homotopy Ft : X×[0,1] → X that continuously deforms X to a point. Thus every cycle c in X is the boundary of some "cone"; one may take the cone to be the image of c under the homotopy. A dual version of this gives Poincaré lemma.
More specifically, we associate to X the cylinder X×[0,1]. Identify the top and bottom of the cylinder with the maps j1(x) = (x, 1) and j0(x) = (x, 0) respectively. On the differential forms, the induced maps j1* and j0* are related by a cochain homotopy K:
Let Ωp(X) denote the p-forms on X. The map K: Ωp + 1( X×[0,1] ) → Ωp(X) is the dual of the cylinder map and defined by
where dxp is a monomial p-form with no dt in it. So if F is a homotopy deforming X to a point Q, then
On forms,
Inserting these two equations into the cochain homotopy equation proves Poincaré lemma.
A corollary of the lemma is that de Rham cohomology is homotopy invariant.
Non-contractible spaces need not have trivial de Rham cohomology. For instance, on the circle S1, parametrized by t in [0, 1], the closed 1-form dt is not exact.
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- This page was last modified on 14 November 2008, at 14:04.
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