Closed and exact differential forms

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

${displaystyle dalpha =0}$

for a given form α to be a closed form, and

${displaystyle alpha =dbeta }$

for an exact form, with ${displaystyle scriptstyle alpha }$ given and ${displaystyle scriptstyle beta }$ unknown.

Like ${displaystyle scriptstyle d^{2}=0}$, being exact is condition enough to be closed. In abstract terms, the main interest of this pair of definitions is to ask if this is also a condition necessary is a way to detect topological information by differential conditions. It doesn't make any real sense to ask if a 0-form is accurate, since d increase the grade to 1.

Overview

Cases of differential forms ${displaystyle scriptstyle mathbb {R} ^{2}}$ and ${displaystyle scriptstyle mathbb {R} ^{3}}$ were already well known in the mathematical physics of the centuryXIX. In the plane, 0-forms are simply functions, and the 2-forms are functions by the basic area element ${displaystyle scriptstyle dxland dy}$So it's the 1-forms

${displaystyle alpha =f(x,y)dx+g(x,y)dy}$

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

${displaystyle dalpha =({g'}_{x}-{f'}_{y})dxland dy}$

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

${displaystyle {f}_{y}={g'}_{x}}}$

In this case if ${displaystyle scriptstyle h(x,y)}$ It's a function then.

${displaystyle dh=h_{x}dx+h_{y}dy}$

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

Poincaré's lemma

The fundamental topological result here is Poincaré's lemma. States that for a contractible open subset of X, any p-differentiable form defined in X that is closed is also exact, for any number integer p > 0 (this has content only when p is at most n).

This is not true for a ring open in the plane, for some 1-forms that do not extend smoothly to the entire disk; so that a certain 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 not exact).