Coarea formula

Remarks

The level sets $u^{-1}(y)$ 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 $\R^n \to \R^n$.

Going deeper

For a student

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.