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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}^m$, and let $a \\in U$.\nSuppose that $f$ is [[differentiable|0135XZZQ8M]] at each point of $U$, and that the derivative $Df \\colon U \\to L(\\mathbb{R}^n, \\mathbb{R}^m)$ is differentiable at $a$.\nThen the second derivative $D^2 f(a)$ is a symmetric bilinear map: $D^2 f(a)(u, v) = D^2 f(a)(v, u)$ for all $u, v \\in \\mathbb{R}^n$.\nIn particular, $\\dfrac{\\partial^2 f}{\\partial x_i \\, \\partial x_j}(a) = \\dfrac{\\partial^2 f}{\\partial x_j \\, \\partial x_i}(a)$ for all $i, j$.\n\n## Remarks\n\nHere $D^2 f(a)$ is the [[iterated derivative|0EY05Z9RPN]] of order $2$, and $D^2 f(a)(e_i, e_j)$ is the mixed [[partial derivative|02RSXVM1ZN]] $\\partial_i \\partial_j f(a)$.\nThe hypothesis is that $f$ is twice differentiable at $a$. It holds when the second partial derivatives are continuous near $a$, which gives [[Schwarz's theorem|0NNQY8WCW6]].\nThe existence of the second partial derivatives at $a$ is not enough: there is a [[function with unequal mixed partial derivatives|01NN8NY497]].\n\n## History\n\nEuler published a proposed proof in 1740, and many incomplete proofs followed over more than a century.\nIn 1867 Lindelöf analysed the earlier proofs and gave a counterexample to the general claim.\nSchwarz gave the first rigorous proof six years later, and Jordan gave in 1883 the general version that most textbooks use.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that the order of two differentiations does not matter, when $f$ is regular enough.\nFor $f(x, y) = x^3 y^2$, differentiating first in $x$ and then in $y$ gives $6 x^2 y$, and the other order gives $6 x^2 y$ too.\n",
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