Namespaces
Variants
Actions

Area formula

From Encyclopedia of Mathematics
Jump to: navigation, search

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

The area formula is a tool to compute the -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

[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe] H. Federer, "Geometric measure theory", Springer-Verlag (1979). MR0257325 Zbl 0874.49001
[Ma] P. Mattila, "Geometry of sets and measures in euclidean spaces". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[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
[Sp] M. Spivak, "Calculus on manifolds" , Benjamin/Cummings (1965) MR0209411 Zbl 0141.05403
How to Cite This Entry:
Area formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Area_formula&oldid=30707