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  "body": "\n## Definition\n\nLet $U, V \\subseteq \\mathbb{R}^n$ be open, and let $k \\ge 1$ be an integer or $k = \\infty$.\nA map $f \\colon U \\to V$ is a $C^k$ diffeomorphism if $f$ is a bijection of [[class C^k|0TCBZMEYGN]] on $U$. In addition, $f^{-1} \\colon V \\to U$ must be of class $C^k$ on $V$.\nThe sets $U$ and $V$ are $C^k$ diffeomorphic if there is a $C^k$ diffeomorphism from $U$ to $V$.\n\n## Remarks\n\nIf $f$ is a diffeomorphism, the chain rule applied to $f^{-1} \\circ f = \\mathrm{id}$ shows that $Df(x)$ is invertible for each $x \\in U$.\nConversely, a $C^k$ bijection $f \\colon U \\to V$ with $Df(x)$ invertible at each point is a $C^k$ diffeomorphism, by the [[inverse function theorem|0K9HSJFCMJ]].\nThe map $x \\mapsto x^3$ is a $C^\\infty$ bijection of $\\mathbb{R}$, but it is not a diffeomorphism, because its inverse $x \\mapsto x^{1/3}$ is not differentiable at $0$.\nPolar coordinates $(r, \\theta) \\mapsto (r \\cos \\theta, r \\sin \\theta)$ give a $C^\\infty$ diffeomorphism from $(0, \\infty) \\times (0, 2\\pi)$ onto the plane without the closed half-line $\\{(x, 0) : x \\ge 0\\}$.\n\n## Explanations\n\n### L1 human\n\nA diffeomorphism is a smooth change of coordinates that can be undone smoothly.\nIt may stretch and bend space, but it does not tear, fold or crush it.\n",
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