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  "body": "\n## Statement\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $f \\colon M \\to \\R$ be smooth.\nLet $p$ be a nondegenerate [[critical point|0YTCMFB4HC]] of $f$ of index $\\lambda$, in the sense of [[Morse functions|KPQ96640ZJ]].\nThen there is a chart $(U, (x_1, \\dots, x_n))$ of $M$ with $p \\in U$ and $x_i(p) = 0$ for all $i$, such that on $U$\n$$f = f(p) - x_1^2 - \\cdots - x_\\lambda^2 + x_{\\lambda+1}^2 + \\cdots + x_n^2.$$\n\n## Remarks\n\nThe idea of the proof is this.\nWork in a chart centred at $p$ with $f(p) = 0$; by Taylor's formula with integral remainder, $f(x) = \\sum_{i,j} x_i x_j h_{ij}(x)$ with $h_{ij}$ smooth, symmetric in $i, j$, and $h_{ij}(0) = \\frac{1}{2} \\partial^2 f / \\partial x_i \\partial x_j(0)$.\nSince the matrix $(h_{ij}(0))$ is invertible, one can complete squares one variable at a time, as in the diagonalisation of a quadratic form, with smooth coefficients near $0$.\nSylvester's law of inertia shows that the number of minus signs is the index of the [[Hessian|0693Y2METT]].\nExample: $f(x, y) = x^2 - y^2 + y^3$ has a nondegenerate critical point of index $1$ at the origin, and near it the change $u = x$, $v = y \\sqrt{1 - y}$ gives $f = u^2 - v^2$.\nNondegeneracy is needed: $f(x) = x^3$ cannot be written as $\\pm x_1^2$ in any chart near $0$, since it changes sign at $0$.\n\n## Explanations\n\n### L1 human\n\nNear a clean peak, pit or saddle, a function looks exactly like a sum of squares with some minus signs, once one chooses the right curved coordinates.\nThe number of minus signs counts the independent downhill directions.\nSo the local picture at a nondegenerate critical point is completely determined by one integer.\n\n### L2 human\n\nThe Morse lemma is the local input of Morse theory.\nIt shows that nondegenerate critical points are isolated, and it describes how the sublevel set $\\set{f \\le c}$ changes when $c$ passes a critical value: a handle of index $\\lambda$ is attached.\nThis leads to handle decompositions, to the Morse inequalities, and to Morse homology.\nIt is the simplest case of the classification of singularities of smooth functions.\n",
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