Area formula

2020 Mathematics Subject Classification: Primary: 49Q15 [MSN][ZBL]

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'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 $Df|^t_y \cdot Df|_y$ (which, by the Cauchy Binet formula, equals the sum of the squares of the determinants of all $n\times n$ minors of the Jacobian 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 [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}

The area formula can be further generalized to Lipschitz maps defined on $n$-dimensional rectifiable subsets of the Euclidean space after defining appropriately a notion of tangential derivation of the Lipschitz map $f$: we refer the reader to Definition 2.89 and Theorems 2.90 and 2.91 in [AFP].

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 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 submanifold $f(U)$ of $\mathbb R^m$ and $Jf (y) = \sqrt{\det g}$ (cp. with Example D in 3.3.4 of [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 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 \mbox{-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'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

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[Ru]	W. Rudin, "Principles of mathematical analysis", Third edition, McGraw-Hill (1976) MR038502 Zbl 0346.2600
[Si]	L. Simon, "Lectures on geometric measure theory", Proceedings of the Centre for Mathematical Analysis, 3. Australian National University. Canberra (1983) MR0756417 Zbl 0546.49019
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