{
  "aliases": [
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    "jacobian-change-of-variables"
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, and let $\\varphi \\colon U \\to \\mathbb{R}^n$ be injective and of [[class C^1|0TCBZMEYGN]] with $\\det D\\varphi(u) \\ne 0$ for each $u \\in U$.\nLet $f \\colon \\varphi(U) \\to \\mathbb{R}$ be continuous with compact support in $\\varphi(U)$.\nThen\n$$\\int_{\\varphi(U)} f(v) \\, dv = \\int_U f(\\varphi(u)) \\, \\lvert \\det D\\varphi(u) \\rvert \\, du.$$\n\n## Remarks\n\nHere $\\det D\\varphi(u)$ is the [[Jacobian determinant|03W6Q1MFR5]] of $\\varphi$ at $u$. Under these hypotheses, $\\varphi(U)$ is open and $\\varphi$ is a [[diffeomorphism|0Z1Y7JHGBR]] from $U$ onto $\\varphi(U)$.\nThe factor $\\lvert \\det D\\varphi(u) \\rvert$ is the ratio by which $\\varphi$ scales small volumes near $u$. A linear map $A$ maps the unit cube to a parallelepiped of volume $\\lvert \\det A \\rvert$.\nThe formula extends to Lebesgue integrable $f$ on $\\varphi(U)$, and to maps that fail the hypotheses only on a set of measure zero.\nImportant cases are [[integration in polar coordinates|0T89W1QZ9K]], cylindrical coordinates and [[spherical coordinates|05WKRC7HWT]].\n\n## Explanations\n\n### L1 human\n\nIn one variable, substitution needs the factor $\\varphi'(u)$. In several variables, the Jacobian determinant plays this role.\nIt corrects for the way the change of coordinates stretches or squeezes small regions.\n",
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  "msc": [
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  "name": "Change of variables in multiple integrals",
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