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  "body": "\n## Summary\n\nThe Morse index of a [[critical point|0YTCMFB4HC]] is the number of independent directions in which the function decreases to second order.\nIt is the largest dimension of a subspace on which the [[Hessian|0693Y2METT]] is negative definite.\nIn Morse theory, passing a [[nondegenerate critical point|MGPPCDFQDG]] of index $\\lambda$ attaches a cell of dimension $\\lambda$ to the sublevel set.\n\n## Explanations\n\n### L1 human\n\nAt the bottom of a bowl no direction goes down, so the index is $0$.\nAt a mountain pass one direction goes down on each side, so the index is $1$.\nAt the top of a hill in a surface every direction goes down, so the index is $2$.\nFor example, $x^2 - y^2 - z^2$ on $\\R^3$ has a critical point of index $2$ at the origin, since it decreases along the $y$ and $z$ axes.\n",
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