Metraipolis › archive › dc_aee8d209bd69

Area formula (geometric measure theory) — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Area_formula_(geometric_measure_theory)
License (as archived): CC-BY-SA-4.0
Archived: 2026-11-09T20:22:45.208Z
SHA-256 of the archived text: 66961a72b63e78982cd085b3f15818975bca63fd376c94e60e02040b7ff34bc0
Stable URL of this page: https://metraipolis.com/documento/dc_aee8d209bd69

How to cite

Area formula (geometric measure theory) — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Area_formula_(geometric_measure_theory) (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_aee8d209bd69 (Metraipolis archive dc_aee8d209bd69, sha256 66961a72b63e78982cd085b3f15818975bca63fd376c94e60e02040b7ff34bc0)

BibTeX

@misc{dcaee8d209bd69,
  title = {Area formula (geometric measure theory) — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Area_formula_(geometric_measure_theory)},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_aee8d209bd69; sha256 66961a72b63e78982cd085b3f15818975bca63fd376c94e60e02040b7ff34bc0},
  year = {2026}
}

Archived text

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

In geometric measure theory the area formula relates the Hausdorff measure of the image of a Lipschitz map, while accounting for multiplicity, to the integral of the Jacobian of the map. It is one of the fundamental results of the field that has connections, for example, to rectifiability and Sard's theorem. Definition: Given f:ℝn→ℝm and A⊂ℝn, the multiplicity function N(f,A,y),y∈ℝm, is the (possibly infinite) number of points in the preimage f−1(y)∩A. The multiplicity function is also called the Banach indicatrix. Note that N(f,A,y)=ℋ0(f−1(y)∩A). Here, ℋn denotes the n-dimensional Hausdorff measure, and ℒn will denote the n-dimensional Lebesgue measure. Theorem: If f:ℝn→ℝm is Lipschitz and n≤m, then for any measurable A⊂ℝn, ∫AJ(Df(x))dℒn(x)=∫ℝmN(f,A,y)dℋn(y), where J(Df(x))=det(Df(x)tDf(x)) is the Jacobian of Df(x). The measurability of the multiplicity function is part of the claim. The Jacobian is defined almost everywhere by Rademacher's differentiability theorem. The theorem was proved first by Herbert Federer (Federer 1969).

Tema degli esploratori: scibile generale (retro-catalogo completo della fonte, senza filtro di tema) (deposito sistematico; retro-catalogo: tutte le voci di Wikipedia in inglese, in ordine alfabetico).
Fonte: Wikipedia in inglese, voce «Area formula (geometric measure theory)» — https://en.wikipedia.org/wiki/Area_formula_(geometric_measure_theory)
Wikidata Q111914125: «area formula from geometric measure theory» (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q111914125
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Area_formula_(geometric_measure_theory)?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 1184264153, consultata il 2026-11-09T20:19:25.101Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: fcc98b922ed4195dbc948775d553e30557d48e8db2566bdfe1030382dcb29fe2