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  "body": "\n## Summary\n\nThe gradient flow of a smooth function $f$ on a [[Riemannian manifold|7CBT4SM83Y]] is the [[flow|06738GGWWK]] of the vector field $-\\nabla f$.\nSome authors use $\\nabla f$ instead.\nAlong its flow lines the function decreases, strictly away from the [[critical points|0YTCMFB4HC]], which are exactly the rest points of the flow.\nIt is the basic tool of Morse theory and of critical point theory: it deforms sublevel sets of $f$ and connects critical points.\n\n## Explanations\n\n### L1 human\n\nThink of $f$ as the height of a landscape and let water run downhill along the steepest path.\nThe path that a drop follows is a flow line of the negative gradient flow.\nDrops stop only where the ground is flat, that is, at critical points of $f$.\nFor example, for $f(x, y) = x^2 + y^2$ on the plane, the flow lines are the rays $t \\mapsto e^{-2t}(x_0, y_0)$, which run into the minimum at the origin.\n",
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