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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $k \\ge 1$ be an integer, and let $f \\colon U \\to \\mathbb{R}$ be of [[class C^k|0TCBZMEYGN]] on $U$.\nLet $a \\in U$.\nFor a [[multi-index|0PJXBR6QA3]] $\\alpha = (\\alpha_1, \\dots, \\alpha_n)$, write $|\\alpha| = \\alpha_1 + \\dots + \\alpha_n$, $\\alpha! = \\alpha_1! \\cdots \\alpha_n!$, $h^\\alpha = h_1^{\\alpha_1} \\cdots h_n^{\\alpha_n}$, and $D^\\alpha f = \\partial^{|\\alpha|} f / \\partial x_1^{\\alpha_1} \\cdots \\partial x_n^{\\alpha_n}$.\nThen\n$$f(a + h) = \\sum_{|\\alpha| \\le k} \\frac{D^\\alpha f(a)}{\\alpha!} \\, h^\\alpha + R_k(h), \\qquad \\lim_{h \\to 0} \\frac{R_k(h)}{\\|h\\|^k} = 0.$$\nIf $f$ is of class $C^{k+1}$ and the segment from $a$ to $a + h$ lies in $U$, then\n$$R_k(h) = \\sum_{|\\beta| = k + 1} \\frac{k + 1}{\\beta!} \\, h^\\beta \\int_0^1 (1 - t)^k \\, D^\\beta f(a + t h) \\, dt.$$\n\n## Remarks\n\nThe sum over $|\\alpha| \\le k$ is the Taylor polynomial of order $k$ of $f$ at $a$.\nFor $k = 1$ the statement says that $f$ is differentiable at $a$, with $f(a + h) = f(a) + \\nabla f(a) \\cdot h + o(\\|h\\|)$.\nOne proof applies the one-variable Taylor theorem to $g(t) = f(a + t h)$ on $[0, 1]$, together with the chain rule.\nThe case $k = 2$ is the [[second-order Taylor expansion|0P30YYZN85]].\n\n## Explanations\n\n### L1 human\n\nNear a point, a smooth function of several variables looks like a polynomial in the displacement $h$.\nThe coefficients of that polynomial are the partial derivatives at the point, divided by factorials.\nThe error is smaller than $\\|h\\|^k$, so the approximation gets better quickly as $h$ shrinks.\n",
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        "url": "https://en.wikipedia.org/wiki/Taylor%27s_theorem"
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        "locator": "Derivation for the remainder of multivariate Taylor polynomials",
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