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  "body": "\n## Summary\n\nThe Hodge star operator is a linear map that sends $k$-forms to $(n-k)$-forms.\nIt is defined on an oriented $n$-dimensional inner product space and, pointwise, on an oriented [[Riemannian manifold|7CBT4SM83Y]].\nIt sends a form to its \"orthogonal complement\", using the metric and the orientation.\nWith it one defines the codifferential, the Hodge Laplacian and harmonic forms, and it underlies Hodge theory and Poincaré duality for de Rham cohomology.\n\n## Explanations\n\n### L1 human\n\nIn three-dimensional space, a small oriented plane piece and the arrow perpendicular to it carry the same information.\nThe Hodge star makes this exchange: it turns the plane spanned by $dx$ and $dy$ into the direction $dz$, and back.\nIn $n$ dimensions it exchanges $k$-dimensional pieces with $(n-k)$-dimensional ones.\nFor example, the cross product of two vectors in $\\R^3$ is the Hodge star of their wedge product.\n",
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