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  "body": "\n## Summary\n\nOn a closed Riemannian manifold the [[Laplace–Beltrami operator|F4NWZNTRMD]] has a discrete spectrum $0 = \\lambda_0 \u003c \\lambda_1 \\le \\lambda_2 \\le \\cdots \\to \\infty$ of eigenvalues of finite multiplicity.\nThe eigenfunctions are smooth and form an orthonormal basis of $L^2(M)$.\nThis sequence is the basic object of [[spectral geometry|QKAFRDPGE4]]: it controls the heat equation, the wave equation and the vibrations of the manifold.\n\n## Explanations\n\n### L1 human\n\nThink of the manifold as a drum without boundary.\nIts pure tones are the eigenfunctions of the Laplacian, and the squares of their frequencies are the eigenvalues.\nThe lowest eigenvalue is $0$, for the constant function, which does not vibrate.\nOn the circle of length $2\\pi$ the eigenfunctions are $\\cos k\\theta$ and $\\sin k\\theta$, with eigenvalues $k^2$.\n",
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