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Closed and exact differential forms — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-31T14:32:00.827Z
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Closed and exact differential forms — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_098a3bfbb672 (Metraipolis archive dc_098a3bfbb672, sha256 8a378fbd663f7db937e8194d34f1e61d217578c5a9eefb2a13a65d4c895f60d0)

BibTeX

@misc{dc098a3bfbb672,
  title = {Closed and exact differential forms — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_098a3bfbb672; sha256 8a378fbd663f7db937e8194d34f1e61d217578c5a9eefb2a13a65d4c895f60d0},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0); and an exact form is a differential form, α, that is the exterior derivative of another differential form β, i.e. α = dβ. Thus, an exact form is in the image of d, and a closed form is in the kernel of d (also known as null space). For an exact form α, α = dβ for some differential form β of degree one less than that of α. The form β is called a "potential form" or "primitive" for α. Since the exterior derivative of a closed form is zero, β is not unique, but can be modified by the addition of any closed form of degree one less than that of α. Because d2 = 0, every exact form is necessarily closed. The question of whether every closed form is exact depends on the topology of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma. More general questions of this kind on an arbitrary differentiable manifold are the subject of de Rham cohomology, which allows one to obtain purely topological information using differential methods.

Tema degli esploratori: fisica e astronomia (deposito sistematico; ricerca «thermodynamics»).
Fonte: Wikipedia in inglese, voce «Closed and exact differential forms» — https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms
Wikidata Q2100733 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q2100733
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1369312779, consultata il 2026-10-31T14:20:32.482Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 925216b48dafa2288287c9c6f0e10ea9b52dba6a88524fa271f99380ea98bee4