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  "body": "\n## Summary\n\nThe Lie derivative measures how a [[tensor field|WX302T93TW]] changes along the flow of a vector field $X$.\nIt is defined without any extra structure such as a metric or a connection.\nOn functions it is the directional derivative, and on vector fields it is the [[Lie bracket|68FGQEBD6J]].\nA tensor field is invariant under the flow of $X$ exactly when its Lie derivative along $X$ is zero.\n\n## Explanations\n\n### L1 human\n\nImagine a fluid flowing on a surface, and some quantity painted on the surface, such as a field of arrows.\nCarry the painted field back along the flow for a short time and compare it with the field that is already there.\nThe rate of change of this comparison is the Lie derivative.\nFor the rotation field $X = -y \\, \\partial_x + x \\, \\partial_y$ in the plane and the function $f = x^2 + y^2$, one gets $X f = 0$.\nThis says that $f$ does not change under rotations about the origin.\n",
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