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  "body": "\n## Summary\n\nA multi-index of dimension $n$ is an $n$-tuple $\\alpha = (\\alpha_1, \\dots, \\alpha_n)$ of non-negative integers.\nWith $|\\alpha| = \\alpha_1 + \\dots + \\alpha_n$, $\\alpha! = \\alpha_1! \\cdots \\alpha_n!$, $x^\\alpha = x_1^{\\alpha_1} \\cdots x_n^{\\alpha_n}$ and $\\partial^\\alpha = \\partial_1^{\\alpha_1} \\cdots \\partial_n^{\\alpha_n}$, formulas of several variables take the short form of their one-variable versions.\n\n## Remarks\n\nHere $\\partial_i$ is the [[partial derivative|02RSXVM1ZN]] with respect to $x_i$, and $\\partial^\\alpha f$ is a partial derivative of order $|\\alpha|$.\nFor $\\alpha = (2, 0, 1)$, $|\\alpha| = 3$, $\\alpha! = 2$, $x^\\alpha = x_1^2 x_3$ and $\\partial^\\alpha f = \\partial^3 f / \\partial x_1^2 \\, \\partial x_3$.\nThe notation $\\partial^\\alpha f$ does not show the order of differentiation. It is unambiguous when $f$ is of class $C^{|\\alpha|}$, by [[Schwarz's theorem|0NNQY8WCW6]].\nExamples of use are [[Taylor's theorem in several variables|0J8T4K4YTE]] and the multinomial theorem $(x_1 + \\dots + x_n)^k = \\sum_{|\\alpha| = k} \\frac{k!}{\\alpha!} x^\\alpha$.\n\n## Explanations\n\n### L1 human\n\nMulti-index notation is a bookkeeping device.\nOne symbol $\\alpha$ records how many times each variable appears, so one formula covers all mixed derivatives or monomials of a given order.\n",
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