View source for Area formula (geometric measure theory)

{{Short description|Area formula from geometric measure theory}}

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 matrix and determinant|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 <math>f\colon \mathbb{R}^n \to \mathbb{R}^m</math> and <math>A\subset \mathbb{R}^n </math>, the multiplicity function <math>N(f,A,y), \, y\in \mathbb{R}^m </math>, is the (possibly infinite) number of points in the preimage <math>f^{-1}(y)\cap A</math>. The multiplicity function is also called the Banach indicatrix. Note that <math>N(f,A,y) = \mathcal{H}^0(f^{-1}(y)\cap A)</math>. Here, <math>\mathcal{H}^n</math> denotes the ''n''-dimensional [[Hausdorff measure]], and <math>\mathcal{L}^n</math> will denote the ''n''-dimensional [[Lebesgue measure]].

Theorem: If <math>f\colon \mathbb{R}^n \to \mathbb{R}^m</math> is Lipschitz and <math>n\leq m</math>, then for any measurable <math>A\subset \mathbb{R}^n</math>,
<math display="block">\int_A {J}(Df(x))\,  d \mathcal{L}^n(x) = \int_{\mathbb{R}^m} N(f,A,y) \, d\mathcal{H}^n(y)  \, ,</math>
where
<math display="block">{J}(Df(x))=\sqrt{\det(Df(x)^tDf(x))}</math>
is the Jacobian of <math>Df(x)</math>.

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]] {{harv|Federer|1969}}.

==Sources==
{{refbegin}}
* {{cite book|mr=1857292|last1=Ambrosio|first1=Luigi|last2=Fusco|first2=Nicola|last3=Pallara|first3=Diego|title=Functions of bounded variation and free discontinuity problems|series=Oxford Mathematical Monographs|publisher=[[Oxford University Press|The Clarendon Press]]|location=New York|year=2000|isbn=0-19-850245-1|author-link1=Luigi Ambrosio|zbl=0957.49001|author-link2=Nicola Fusco}}
* {{cite book|mr=3409135|last1=Evans|first1=Lawrence C.|last2=Gariepy|first2=Ronald F.|title=Measure theory and fine properties of functions|edition=Revised edition of 1992 original|series=Textbooks in Mathematics|publisher=[[CRC Press]]|location=Boca Raton, FL|year=2015|isbn=978-1-4822-4238-6|author-link1=Lawrence Evans|doi=10.1201/b18333|zbl=1310.28001}}
* {{Cite book| last = Federer| first = Herbert| author-link = Herbert Federer| title = Geometric measure theory| place= Berlin–Heidelberg–New York| publisher = [[Springer-Verlag]]| series = Die Grundlehren der mathematischen Wissenschaften| volume = 153| year = 1969| isbn = 978-3-540-60656-7| mr=0257325| zbl= 0176.00801 |doi=10.1007/978-3-642-62010-2}}
* {{cite book|mr=0756417|author-link1=Leon Simon|last1=Simon|first1=Leon|title=Lectures on geometric measure theory|series=Proceedings of the Centre for Mathematical Analysis, Australian National University|volume=3|publisher=Australian National University, Centre for Mathematical Analysis|location=Canberra|year=1983|isbn=0-86784-429-9|url=https://web.stanford.edu/class/math285/ts-gmt.pdf|zbl=0546.49019}}
{{refend}}

==External links==
*{{springer|title=Area formula}}

[[Category:Theorems in measure theory]]