Closed and exact differential forms

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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

\zeta - \eta = d\beta\,

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

\alpha = f(x,y)dx + g(x,y)dy\,

that are of real interest. The formula for the exterior derivative d here is

d \alpha = (g_x - f_y)dx\wedge dy\,

where the subscripts denote partial derivatives. Therefore the condition for α to be closed is

f_y=g_x.\,

In this case if h(x,y) is a function then

dh = h_x dx + h_y dy.\,

The implication from 'exact' to 'closed' is then a consequence of the symmetry of second derivatives, with respect to x and y.

[edit] 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 specificly, 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:

K d + d K = j_1^* - j_0 ^*.

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

a(x,t) d x^{p+1} \mapsto 0, \; a(x,t) dt dx^p \mapsto (\int_0 ^1 a(x,t) dt) dx^p,

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

F \circ j_1 = id, \; F \circ j_0 = Q.

On forms,

j_1 ^* \circ F^* = id, \; j_0^* \circ F^* = 0.

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.