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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $f \\colon M \\to \\R$ be a [[smooth function|0V33BNKZ9S]].\nThe [[gradient|0ABK9V9Y64]] of $f$ is the vector field $\\nabla f$ on $M$ with $g(\\nabla f, X) = df(X)$ for every tangent vector $X$.\nThe *(negative) gradient flow* of $f$ is the [[flow|06738GGWWK]] $\\varphi$ of $-\\nabla f$.\nSo it is the solution of\n$$\\frac{\\partial}{\\partial t} \\varphi_t(x) = -\\nabla f\\big(\\varphi_t(x)\\big), \\qquad \\varphi_0(x) = x,$$\ndefined on its maximal domain.\n\n## Remarks\n\nAlong a flow line, $\\frac{d}{dt} f(\\varphi_t(x)) = -\\abs{\\nabla f}^2 \\le 0$, with equality exactly at [[critical points|0YTCMFB4HC]] of $f$.\nSo $f$ is a Lyapunov function for the flow, and the rest points of the flow are the critical points of $f$.\nIf $M$ is compact, the flow is defined for all $t \\in \\R$.\n\nExample: on $\\R^2$ with the Euclidean metric, let $f(x, y) = x^2 - y^2$.\nThen $-\\nabla f = (-2x, 2y)$ and $\\varphi_t(x, y) = (e^{-2t} x, e^{2t} y)$.\nThe origin is the only rest point, and the flow has the familiar saddle picture.\n\nNon-example: on $\\R^2$, the rotation field $X(x, y) = (-y, x)$ is not $-\\nabla f$ for any smooth $f$.\nIts flow has closed orbits, but $f$ would strictly decrease along a nonconstant flow line of a gradient flow, so it could not return to its starting point.\n\nThe flow depends on the metric $g$, not only on $f$.\nA vector field $X$ with $df(X) \u003c 0$ away from the critical points, and of a standard form near them, is called gradient-like for $f$.\nSuch fields serve the same purposes in Morse theory.\nFor a Morse function and a generic metric, the gradient flow is a [[Morse–Smale system|7AMQ8GS78Y]].\n",
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