*d*α = 0

and

- α =
*d*β

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

The cases of differential forms in **R**^{2} and **R**^{3} 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*

*d*α = (*f*_{y}−*g*_{x})*dx*.*dy*

*f*_{y}=*g*_{x}.

*dh*=*h*_{x}*dx*+*h*_{y}*dy*.

The fundamental topological result here is the **Poincaré lemma**. It states that for a contractible open subset *X* of **R**^{n}, 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).