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Closed and exact differential forms

In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations

dα = 0

for a given form α to be a closed form,


α = dβ

for an exact form, with α given and β unknown.

Since d2 = 0, to be exact is a sufficient condition to be closed. In abstract terms, the main interest of this pair of definitions 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.

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

α = f(x,y)dx + g(x,y)dy

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

dα = (fygx)dx.dy

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

fy = gx.

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

dh = hxdx + hydy.

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

The fundamental topological result here is the Poincaré lemma. It states that for a contractible open subset X of Rn, any smooth p-form α defined on X that is closed, is also exact, for any integer p > 0 (this has content only when p is at most n).

This is not true for an open annulus in the plane, for some 1-forms α that fail to extend smoothly to the whole disk; so that some topological condition is necessary.

In terms of De Rham cohomology, the lemma says that contractible sets have the cohomology groups of a point (considering that the constant 0-forms are closed but vacuously aren't exact).