View source for Area formula


{{MSC|49Q15}}

[[Category:Classical measure theory]]

{{TEX|done}}

The area formula is a tool to compute the $n$-dimensional volume of (sufficiently regular) subsets of the Euclidean space.
In what follows we denote by $\mathcal{H}^n$ the $n$-dimensional [[Hausdorff measure]] in $\mathbb R^m$ and by $\lambda$ the [[Lebesgue measure]] in $\mathbb R^n$. We also recall that $\mathcal{H}^0 (S)$ is the cardinality of the set $S$.

===General statement===
Consider a Lipschitz map $f: \mathbb R^n \to \mathbb R^m$, where $m\geq n$. Recall that, by [[Rademacher theorem|Rademacher's theorem]], $f$ is differentiable $\lambda$-a.e.. At any point $y\in \mathbb R^n$ of differentiability we denote by $J f (y)$ the [[Jacobian]] of $f$ in $y$, that is the square root of the determinant of $Df|_y^t \cdot Df|_y$ (which, by the [[Cauchy Binet formula]], equals the sum of the squares of the determinants of all $n\times n$
minors of the [[Jacobian|Jacobian matrix]] $Df|_y$, see [[Jacobian]]). 

'''Theorem 1'''
The map $y\mapsto Jf (y)$ is Lebesgue measurable. For any Lebesgue measurable set $A\subset \mathbb R^n$ the map
\[
z\mapsto \mathcal{f}^{-1} (\{z\})
\]
is $\mathcal{H}^n$-measurable 
and the following identity holds:
\begin{equation}\label{e:area_formula}
\int_A J f (y)\, dy = \int_{\mathbb R^m} \mathcal{H}^0 (A\cap f^{-1} (\{z\}))\, d\mathcal{H}^n (z)\, .
\end{equation}

Cp. with 3.2.2 of {{Cite|EG}}. From \eqref{e:area_formula} it is not difficult to conclude the following generalization (which also goes often under the same name):

'''Theorem 2'''
For any $\lambda$-summable function $g:\mathbb R^n\to \mathbb R$ the map
\[
z\mapsto \sum_{x\in \mathcal{f}^{-1} (\{z\})} g (x)
\]
is $\mathcal{H}^n$-measurable and
\begin{equation}\label{e:area_formula2}
\int_{\R^n} g (y)\, J f (y)\, dy = \int_{\mathbb R^m} \sum_{x\in \mathcal{f}^{-1} (\{z\})} g (x)\;\,d\mathcal{H}^n (z)\, .
\end{equation}

===Corollaries===
====Change of variables====
If $n=m$ then $\mathcal{H}^n$ coincides with the Lebesgue measure $\lambda$ on $\mathbb R^n$.
Assume in addition that $f:\mathbb R^n \to \mathbb R^n$ is injective.
We then conclude from \eqref{e:area_formula2}:
\[
\int_{\R^n} g(y)\, Jf (y)\, dy = \int_{f (\mathbb R^n)} g (f^{-1} (z))\, dz\, .
\]
Thus

'''Corollary 3'''
Assume $f:\mathbb R^n \to \mathbb R^n$ is an injective Lipschitz map and $h$ a $\lambda$-summable function. Then
\begin{equation}\label{e:change_of_var}
\int_{\R^n} h(f(y))\, Jf (y)\, dy = \int_{f (\mathbb R^n)} h(z)\, dz\, .
\end{equation}

This statement generalizes the usual [[Change of variables in an integral|change of variables formula]] for $n$-dimensional integrals.
====The volume of submanifolds====
If $f: \mathbb R^n\to \mathbb R^m$ is a Lipschitz injective map and $A\subset \mathbb R^n$ a $\lambda$-measurable subset of $\mathbb R^n$ with finite measure, we then conclude from \eqref{e:area_formula} that $f (E)$ is $\mathcal{H}^n$-measurable and 
\[
\mathcal{H}^n (f(E)) = \int_E J f(y)\, dy\, .
\]
Assume next that $f$ is $C^1$ and let $U$ be an open set of $\mathbb R^n$ on which the differential of $f$ has maximum rank everywhere. If we set
\[
g_{ij} = \partial_i f \cdot \partial_j f\, ,
\]
then $g$ is the metric tensor of the [[Riemannian manifold|Riemannian submanifold]] $f(U)$ of $\mathbb R^m$ and $Jf (y) = \sqrt{\det g}$ (cp. with Example D in 3.3.4 of {{Cite|EG}}). We therefore conclude

'''Corollary 4'''
Assume that $\Sigma\subset \R^m$ is an $n$-dimensional $C^1$ submanifold. Then the [[Hausdorff dimension]] of $\Sigma$ is $n$ and, for any relatively open set $U\subset \Sigma$,
\[
\mathcal{H}^n (U) = \int_U {\rm d\, vol}
\]
where ${\rm d\, vol}$ denotes the usual [[Volume form|volume form]] of $\Sigma$ as Riemannian submanifold of $\mathbb R^n$.
====Sard's type statements====
If $L$ denotes the Lipschitz constant of $f$, it follows easily that $|Jf|\leq L^n$ $\lambda$-a.e..
Thus, from \eqref{e:area_formula} we conclude

'''Corollary 5'''
Let $f: \mathbb R^n \to \mathbb R^m$ be a Lipschitz map with $m\geq n$ and assume that $A\subset \mathbb R^n$ is a Lebesgue measurable set with finite measure. Then
\begin{equation}\label{e:Sard}
A\cap f^{-1} (\{z\}) \mbox{ is a finite set for $\mathcal{H}^n$-a.e. $z\in \mathbb R^m$.}
\end{equation}

When $n=m$, $f$ is $C^1$ and $A$ is a compact set, \eqref{e:Sard} is a corollary of [[Sard theorem|Sard's theorem]]: indeed, according to Sard's theorem, $\lambda$-a.e. $z\in \mathbb R^n$ is a regular value. For such $z$, $f^{-1} (\{z\})$ is a discrete set and therefore $A\cap f^{-1} (\{z\})$ is finite. For this reason Corollary 5 can be considered a generalization of Sard's theorem to Lipschitz maps $f: \mathbb R^n \to \mathbb R^m$ when $m\geq n$. An analogous statement in the case $m<n$ can be inferred from the [[Coarea formula]].

===References===
{|
|-
|valign="top"|{{Ref|EG}}||  L.C. Evans, R.F. Gariepy,  "Measure theory   and fine properties of  functions" Studies in Advanced  Mathematics.  CRC  Press, Boca Raton,  FL,  1992. {{MR|1158660}}  {{ZBL|0804.2800}} 
|-
|valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure theory", Springer-Verlag (1979). {{MR|0257325}} {{ZBL|0874.49001}}
|-
|valign="top"|{{Ref|Ma}}||      P. Mattila, "Geometry of sets and measures in euclidean spaces".      Cambridge Studies in Advanced Mathematics, 44. Cambridge University      Press, Cambridge,  1995. {{MR|1333890}} {{ZBL|0911.28005}}
|-
|valign="top"|{{Ref|Ru}}||  W. Rudin, "Principles of  mathematical  analysis", Third edition,  McGraw-Hill (1976) {{MR|038502}}   {{ZBL|0346.2600}}  
|-
|valign="top"|{{Ref|Si}}||  L. Simon, "Lectures on geometric measure theory", Proceedings of the  Centre for Mathematical Analysis, 3. Australian National University.  Canberra (1983) {{MR|0756417}} {{ZBL|0546.49019}} 
|-
|valign="top"|{{Ref|Sp}}|| M. Spivak,   "Calculus on manifolds" , Benjamin/Cummings  (1965) {{MR|0209411}} {{ZBL|0141.05403}}
|-
|}