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  "body": "\n## Summary\n\nThe Laplace–Beltrami operator is the [[Laplacian|0E0B6RA8K0]] of a [[Riemannian manifold|7CBT4SM83Y]]: the divergence of the gradient of a function, both taken with respect to the metric.\nIt is a second-order elliptic operator that depends only on the metric, not on coordinates.\nIts eigenvalues and eigenfunctions form the subject of spectral geometry, and it governs heat flow, waves and harmonic functions on manifolds.\n\n## Explanations\n\n### L1 human\n\nOn flat space, the Laplacian of a function compares its value at a point with its average on small spheres around that point.\nThe Laplace–Beltrami operator does the same thing on a curved space, measuring distances with the curved metric.\nOn the unit sphere, the function giving the height above the equator is an eigenfunction: its Laplacian is $-2$ times itself.\nThe vibrations of a curved membrane and the way heat spreads on a curved surface are both described by this operator.\n",
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