Exact differential form

Let $M$ be a smooth manifold . Exactness is defined using the exterior derivative on differential forms .

Definition

A form $\omega\in\Omega^k(M)$ is exact if there exists $\eta\in\Omega^{k-1}(M)$ such that

The vector space of exact $k$-forms is

Exact forms are automatically closed because $d^2=0$. In the de Rham cohomology group $H^k_{\mathrm{dR}}(M)$, exact forms represent the zero class.

Examples

1. Differentials of functions are exact 1-forms.
   For any smooth function $f\in C^\infty(M)$, the 1-form $df$ is exact by definition, with $\eta=f$ viewed as a 0-form.

2. A basic exact 2-form on $\mathbb{R}^3$.
   On $\mathbb{R}^3$ with coordinates $(x,y,z)$,

   $$dx\wedge dy = d(x\,dy),$$

   so $dx\wedge dy$ is exact.

3. A closed non-example (not exact globally).
   On $U=\mathbb{R}^2\setminus\{0\}$, the 1-form

   $$\omega=\frac{-y\,dx + x\,dy}{x^2+y^2}$$

   is closed but not exact; equivalently, it defines a nonzero class in the de Rham cohomology of $U$.