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  "body": "\n## Example\n\nLet $S^n \\subset \\R^{n+1}$ be the unit [[sphere|P9CM4GQ2T0]] with its round metric, $n \\ge 1$, and let $\\Delta = -\\operatorname{div} \\operatorname{grad}$ be its [[Laplace–Beltrami operator|F4NWZNTRMD]].\nIts eigenvalues are\n$$\\lambda_k = k(k + n - 1), \\quad k = 0, 1, 2, \\ldots.$$\nThe $\\lambda_k$-eigenspace consists of the restrictions to $S^n$ of the homogeneous harmonic polynomials of degree $k$ on $\\R^{n+1}$, the [[spherical harmonics|3F32Y21HP4]] of degree $k$.\nIts dimension is\n$$\\binom{n + k}{n} - \\binom{n + k - 2}{n},$$\nwith the convention that the second term is $0$ when $k \u003c 2$.\n\nCheck of the eigenvalue: write a homogeneous harmonic polynomial of degree $k$ as $P = r^k Y$ with $r = \\abs{x}$ and $Y$ a function on $S^n$.\nIn polar coordinates the Euclidean Laplacian is $\\partial_r^2 + \\frac{n}{r} \\partial_r - \\frac{1}{r^2} \\Delta$.\nApplied to $P$ it gives $r^{k-2} \\big( k(k-1) + nk \\big) Y - r^{k-2} \\Delta Y = 0$, so $\\Delta Y = k(k + n - 1) Y$.\nThese eigenfunctions span a dense subspace of $L^2(S^n)$, so there are no other eigenvalues.\n\nFor $S^2$ the eigenvalues are $k(k+1)$ with multiplicity $2k + 1$.\nFor $k = 1$ the eigenfunctions are the restrictions of the coordinate functions $x_1, \\ldots, x_{n+1}$, with eigenvalue $n$.\n\n## Remarks\n\nSince the unit sphere has Ricci curvature $(n - 1)$ times the metric, the first nonzero eigenvalue $n$ shows that the bound in the [[Lichnerowicz–Obata theorem|TK1PT9FDV0]] is attained.\nThe multiplicities grow like a constant times $k^{n-1}$, which agrees with [[Weyl's law|2JK9YMZV2B]].\n\n## Explanations\n\n### L1 human\n\nThe sphere vibrates only at a discrete list of frequencies, and the list can be written down exactly.\nThe $k$-th frequency comes from polynomials of degree $k$, such as $x$ for $k = 1$ or $xy$ for $k = 2$.\nOn the ordinary sphere $S^2$ the values are $0, 2, 6, 12, \\ldots$, with $1, 3, 5, 7, \\ldots$ independent vibration patterns.\nThese patterns are the spherical harmonics used for orbitals in chemistry and for maps of the Earth's gravity.\n",
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