{
  "aliases": [
    "second-order-taylor-formula"
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}$ be of [[class C^2|0TCBZMEYGN]] on $U$, and let $a \\in U$.\nLet $\\nabla f(a)$ be the [[gradient|0NS2QYVC4Y]] of $f$ at $a$, and let $H_f(a)$ be the [[Hessian matrix|0SCSQWXRA4]] of $f$ at $a$.\nThen\n$$f(a + h) = f(a) + \\nabla f(a) \\cdot h + \\tfrac{1}{2} \\, h^{\\mathsf{T}} H_f(a) \\, h + R(h), \\qquad \\lim_{h \\to 0} \\frac{R(h)}{\\|h\\|^2} = 0.$$\n\n## Remarks\n\nThis is the case $k = 2$ of [[Taylor's theorem in several variables|0J8T4K4YTE]], written with matrices. The quadratic term collects the terms with $|\\alpha| = 2$.\nFor $n = 2$ and $h = (u, v)$, the quadratic term is $\\tfrac{1}{2} \\left( f_{xx}(a) u^2 + 2 f_{xy}(a) u v + f_{yy}(a) v^2 \\right)$.\nAt a critical point the linear term is zero, so the sign of $h^{\\mathsf{T}} H_f(a) h$ controls the behaviour of $f$ near $a$. This is the basis of the [[second derivative test|0GYDV9FE98]].\n\n## Explanations\n\n### L1 human\n\nA function of class $C^2$ is close to a quadratic polynomial near each point.\nThe gradient gives the linear part and the Hessian gives the quadratic part.\nThe error goes to zero faster than $\\|h\\|^2$.\n",
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    "name": "Second-order Taylor expansion",
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        "locator": "Taylor's theorem for multivariate functions",
        "url": "https://en.wikipedia.org/wiki/Taylor%27s_theorem"
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      {
        "locator": "Example in two dimensions",
        "url": "https://en.wikipedia.org/wiki/Taylor%27s_theorem"
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  "markdown": "https://math2.tamnd.com/node/0P30YYZN85.md",
  "msc": [
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  "name": "Second-order Taylor expansion",
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