Knowlpedia

Exact differential form

A differential form that is the exterior derivative of a form of one lower degree.

Let MM be a . Exactness is defined using the on .

For k≥0k\ge0, with the convention Ω−1(M)=0\Omega^{-1}(M)=0, a form ω∈Ωk(M)\omega\in\Omega^k(M) is exact if there exists η∈Ωk−1(M)\eta\in\Omega^{k-1}(M) such that

ω=dη.\omega = d\eta.

The vector space of exact kk-forms is

Bk(M)≔im⁡ ⁣(d:Ωk−1(M)→Ωk(M)).B^k(M) \coloneqq \operatorname{im}\!\bigl(d:\Omega^{k-1}(M)\to\Omega^k(M)\bigr).
Relation to closed forms

Exact forms are automatically because d2=0d^2=0. In the HdRk(M)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∈C∞(M)f\in C^\infty(M), the 1-form dfdf is exact by definition, with η=f\eta=f viewed as a 0-form.
  1. A basic exact 2-form on R3\mathbb{R}^3. On R3\mathbb{R}^3 with coordinates (x,y,z)(x,y,z),
    dx∧dy=d(x dy),dx\wedge dy = d(x\,dy),
    so dx∧dydx\wedge dy is exact.
  1. A closed non-example (not exact globally). On U=R2∖{0}U=\mathbb{R}^2\setminus\{0\}, the 1-form
    ω=−y dx+x dyx2+y2\omega=\frac{-y\,dx + x\,dy}{x^2+y^2}
    is but not exact; equivalently, it defines a nonzero class in the of UU.