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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,
and
- α = dβ
for an exact form, with α given and β unknown.
Since d 2 = 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 = (fy − gx)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).
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