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  "aliases": [
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open and let $f \\colon U \\to \\mathbb{R}$ be of [[class C^2|0TCBZMEYGN]] on $U$.\nThen for all $i, j \\in \\{1, \\dots, n\\}$ and each $a \\in U$,\n$$\\frac{\\partial^2 f}{\\partial x_i \\, \\partial x_j}(a) = \\frac{\\partial^2 f}{\\partial x_j \\, \\partial x_i}(a).$$\n\n## Remarks\n\nA function of class $C^2$ is twice differentiable at each point, so this is a case of the [[symmetry of second derivatives|01V9FPBBVK]].\nThe result is also called Clairaut's theorem. By induction, for a function of class $C^k$ the [[partial derivatives|02RSXVM1ZN]] of order at most $k$ do not depend on the order of differentiation.\nSo for $C^k$ functions one can use [[multi-index notation|0PJXBR6QA3]] $\\partial^\\alpha f$ without ambiguity.\n\n## Explanations\n\n### L1 human\n\nIf the second partial derivatives are continuous, then $f_{xy} = f_{yx}$.\nThis saves work: in a $C^2$ function of $n$ variables, only $n(n+1)/2$ of the $n^2$ second partial derivatives can differ.\n",
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