---
tag: QK55379SXW
kind: statement
role: theorem
name: Coarea formula
msc: [28A75, 26B15, 49Q15]
uses: [0AW1E6YTJY, Q1TGZK8RN6, 2Z2QVWZ262, 047NDTEPCV, 0B2RZ0778N, YYNCJSR0PJ]
about: [Q1TGZK8RN6, 0B2RZ0778N]
sources:
  - url: https://en.wikipedia.org/wiki/Coarea_formula
same_as:
  wikidata: Q2591257
  wikipedia: Coarea formula
---

## Statement

Let $n \ge m \ge 1$ be integers and let $u \colon \R^n \to \R^m$ be [[Lipschitz|0AW1E6YTJY]].
By Rademacher's theorem, $u$ is differentiable at [[Lebesgue|2Z2QVWZ262]]-almost every $x$, with [[Jacobian matrix|047NDTEPCV]] $Du(x)$ of size $m \times n$.
Let $J_m u(x) = \sqrt{\det\big(Du(x) \, Du(x)^T\big)}$ at those points.
Let $\mathcal{H}^{n-m}$ be the $(n-m)$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] on $\R^n$.
Then for every Lebesgue [[measurable|YYNCJSR0PJ]] $g \colon \R^n \to [0, \infty]$,
$$\int_{\R^n} g(x) \, J_m u(x) \, dx = \int_{\R^m} \left( \int_{u^{-1}(y)} g \, d\mathcal{H}^{n-m} \right) dy.$$
The same holds for every Lebesgue integrable $g \colon \R^n \to \R$, the inner integral being defined for almost every $y$.

## Remarks

The [[level sets|0B2RZ0778N]] $u^{-1}(y)$ slice $\R^n$, and the formula says that integrating over $\R^n$ is the same as integrating over each slice and then over the parameter $y$.
The weight $J_m u$ measures how much $u$ squeezes volume across the slices.
For a linear $u$ of full rank, the formula is a computation with Fubini's theorem; the general case uses approximation of a Lipschitz map by $C^1$ maps on large sets.

Example: with $m = 1$ and $u(x) = \abs{x}$, one has $J_1 u = \abs{\nabla u} = 1$ for $x \ne 0$, and the level sets are spheres.
The formula becomes integration in polar coordinates: $\int_{\R^n} g \, dx = \int_0^\infty \int_{\set{\abs{x} = r}} g \, d\mathcal{H}^{n-1} \, dr$.

For $m = 1$ and $g = 1$ on a set $A$, it reads $\int_A \abs{\nabla u} \, dx = \int_{\R} \mathcal{H}^{n-1}(A \cap \set{u = t}) \, dt$.
For $m = n$, $\mathcal{H}^0$ is counting measure and the formula is the area formula for maps $\R^n \to \R^n$.

## Explanations

### L1 human

Cut a solid into thin layers along the level sets of a function, like slicing an onion.
The volume of the solid equals the sum of the areas of the layers, weighted by how far apart neighbouring layers are.
Where the function changes fast, the layers are close together, and the weight $\abs{\nabla u}$ corrects for that.

### L2 human

The coarea formula generalises Fubini's theorem from straight slices to curved ones.
For $m = 1$ and $g = 1$ it links the total variation of a function to the perimeters of its level sets.
This is the starting point of the Fleming–Rishel formula for functions of bounded variation.
It is used to prove isoperimetric and Sobolev inequalities and to compute integrals over level sets in probability and geometry.