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  "body": "\n## Definition\n\nLet $V$ be a real vector space of dimension $n$ with an [[inner product|0NB0AWGJBW]] $\\ip{\\cdot}{\\cdot}$ and an [[orientation|0QJJAQ6ZYZ]].\nLet $\\operatorname{vol} = e_1 \\wedge \\dots \\wedge e_n$ for any positively oriented orthonormal basis $e_1, \\dots, e_n$.\nOn the [[exterior power|0MDMZ965VE]] $\\Lambda^k V$ use the inner product with $\\ip{v_1 \\wedge \\dots \\wedge v_k}{w_1 \\wedge \\dots \\wedge w_k} = \\det\\bigl(\\ip{v_i}{w_j}\\bigr)$.\nThe *Hodge star operator* is the unique linear map $\\star \\colon \\Lambda^k V \\to \\Lambda^{n-k} V$ such that\n$$\\alpha \\wedge \\star\\beta = \\ip{\\alpha}{\\beta} \\operatorname{vol} \\quad \\text{for all } \\alpha, \\beta \\in \\Lambda^k V.$$\nOn an oriented [[Riemannian manifold|7CBT4SM83Y]] $(M, g)$, the Hodge star is applied pointwise to [[differential forms|0E3G0GRPZ6]], with $V = T_p^* M$ and the inner product induced by $g$.\n\n## Remarks\n\nExample: in $\\R^3$ with the standard orientation, $\\star dx = dy \\wedge dz$, $\\star dy = dz \\wedge dx$ and $\\star dz = dx \\wedge dy$.\nCheck for $dx$: $dx \\wedge \\star dx = dx \\wedge dy \\wedge dz = \\ip{dx}{dx} \\operatorname{vol}$, and $dy \\wedge \\star dx = 0 = \\ip{dy}{dx} \\operatorname{vol}$, and the same for $dz$.\n\nNon-example: in $\\R^2$ the map with $dx \\mapsto -dy$ and $dy \\mapsto dx$ is not the Hodge star, since $dx \\wedge (-dy) = -\\operatorname{vol}$, while $\\ip{dx}{dx} \\operatorname{vol} = \\operatorname{vol}$.\nThe Hodge star of $\\R^2$ is $\\star dx = dy$, $\\star dy = -dx$, a rotation by a right angle.\n\nOn orthonormal bases, $\\star(e_{i_1} \\wedge \\dots \\wedge e_{i_k}) = \\pm e_{j_1} \\wedge \\dots \\wedge e_{j_{n-k}}$, where the $j$'s are the complementary indices and the sign is that of the permutation $(i_1, \\dots, i_k, j_1, \\dots, j_{n-k})$.\nIt satisfies $\\star 1 = \\operatorname{vol}$ and $\\star\\star = (-1)^{k(n-k)}$ on $\\Lambda^k V$, and it is an isometry.\nReversing the orientation changes the sign of $\\star$.\n",
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