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  "body": "\n## Statement\n\nLet $n \\ge m \\ge 1$ be integers and let $u \\colon \\R^n \\to \\R^m$ be [[Lipschitz|0AW1E6YTJY]].\nBy Rademacher's theorem, $u$ is differentiable at [[Lebesgue|2Z2QVWZ262]]-almost every $x$, with [[Jacobian matrix|047NDTEPCV]] $Du(x)$ of size $m \\times n$.\nLet $J_m u(x) = \\sqrt{\\det\\big(Du(x) \\, Du(x)^T\\big)}$ at those points.\nLet $\\mathcal{H}^{n-m}$ be the $(n-m)$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] on $\\R^n$.\nThen for every Lebesgue [[measurable|YYNCJSR0PJ]] $g \\colon \\R^n \\to [0, \\infty]$,\n$$\\int_{\\R^n} g(x) \\, J_m u(x) \\, dx = \\int_{\\R^m} \\left( \\int_{u^{-1}(y)} g \\, d\\mathcal{H}^{n-m} \\right) dy.$$\nThe same holds for every Lebesgue integrable $g \\colon \\R^n \\to \\R$, the inner integral being defined for almost every $y$.\n\n## Remarks\n\nThe [[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 $y$.\nThe weight $J_m u$ measures how much $u$ squeezes volume across the slices.\nFor 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.\n\nExample: 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.\nThe formula becomes integration in polar coordinates: $\\int_{\\R^n} g \\, dx = \\int_0^\\infty \\int_{\\set{\\abs{x} = r}} g \\, d\\mathcal{H}^{n-1} \\, dr$.\n\nFor $m = 1$ and $g = 1$ on a set $A$, it reads $\\int_A \\abs{\\nabla u} \\, dx = \\int_{\\R} \\mathcal{H}^{n-1}(A \\cap \\set{u = t}) \\, dt$.\nFor $m = n$, $\\mathcal{H}^0$ is counting measure and the formula is the area formula for maps $\\R^n \\to \\R^n$.\n\n## Explanations\n\n### L1 human\n\nCut a solid into thin layers along the level sets of a function, like slicing an onion.\nThe volume of the solid equals the sum of the areas of the layers, weighted by how far apart neighbouring layers are.\nWhere the function changes fast, the layers are close together, and the weight $\\abs{\\nabla u}$ corrects for that.\n\n### L2 human\n\nThe coarea formula generalises Fubini's theorem from straight slices to curved ones.\nFor $m = 1$ and $g = 1$ it links the total variation of a function to the perimeters of its level sets.\nThis is the starting point of the Fleming–Rishel formula for functions of bounded variation.\nIt is used to prove isoperimetric and Sobolev inequalities and to compute integrals over level sets in probability and geometry.\n",
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