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  "body": "\n## Summary\n\nThe Hodge Laplacian is the operator $\\Delta = d\\delta + \\delta d$ on [[differential forms|0E3G0GRPZ6]] of a Riemannian manifold, built from the [[exterior derivative|0ZHSTAG8ZY]] $d$ and its adjoint $\\delta$.\nOn functions it is the [[Laplace–Beltrami operator|F4NWZNTRMD]], up to sign convention.\nIt is elliptic, self-adjoint and nonnegative, and on a closed manifold its kernel, the harmonic forms, represents the de Rham cohomology.\nThis is the content of [[Hodge theory|32YJXG30MF]], which links analysis on a manifold with its topology.\n\n## Explanations\n\n### L1 human\n\nThe ordinary Laplacian measures how much a function differs from its average nearby.\nThe Hodge Laplacian does the same for differential forms, such as vector fields written as $1$-forms.\nOn flat $\\R^n$ it simply applies the negative of the usual Laplacian $-\\sum_j \\partial_j^2$ to each coefficient of the form.\nOn a curved manifold extra curvature terms appear, which is why it carries geometric information.\nA form killed by it is called harmonic, and on a closed manifold each cohomology class contains exactly one harmonic form.\n",
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