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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$ and let $f \\colon M \\to \\R$ be a [[smooth function|0V33BNKZ9S]].\nLet $p$ be a [[critical point|0YTCMFB4HC]] of $f$, so $df_p = 0$.\nThe *Hessian* of $f$ at $p$ is the [[symmetric bilinear form|0AXZNE5XNH]] on $T_p M$ given by\n$$\\operatorname{Hess}_p f(v, w) = v\\big(W f\\big),$$\nwhere $W$ is any vector field with $W(p) = w$.\nThe critical point $p$ is *nondegenerate* if $\\operatorname{Hess}_p f$ is a [[nondegenerate bilinear form|01NHFZF2SK]].\nOtherwise $p$ is *degenerate*.\n\n## Remarks\n\nBecause $df_p = 0$, the value $v(Wf)$ does not depend on the extension $W$ of $w$, and the form is symmetric.\nIn a chart $(x_1, \\dots, x_n)$ around $p$, $\\operatorname{Hess}_p f$ has the matrix $\\big(\\partial^2 f / \\partial x_i \\partial x_j (p)\\big)$, the [[Hessian matrix|0693Y2METT]], and $p$ is nondegenerate if and only if this matrix is invertible.\n\nExample: $f(x, y) = x^2 + xy + y^2$ on $\\R^2$ has the critical point $(0, 0)$ with Hessian matrix $\\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$.\nIts determinant is $3 \\ne 0$, so the origin is nondegenerate.\n\nNon-example: $g(x, y) = x^2 + y^4$ has the critical point $(0, 0)$ with Hessian matrix $\\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix}$, which is singular, so the origin is degenerate.\n\nNondegenerate critical points are isolated among critical points.\nA function all of whose critical points are nondegenerate is a [[Morse function|KPQ96640ZJ]].\n",
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