Closed and exact differential forms

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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 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.

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

α = f(x,y)dx + g(x,y)dy

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

dα = (g_x − f_y)dx∧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_xdx + h_ydy.

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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