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Hodge star operator∗ rspuzio† 2013-03-21 15:48:39 Let V be a n-dimensional (n finite) vector space with inner product g. The Hodge star operator (denoted by ∗) is a linear operator mapping p-forms on V to (n − p)-forms, i.e., ∗ : Ωp (V ) → Ωn−p (V ). In terms of a basis {e1 , . . . , en } for V and the corresponding dual basis {e1 , . . . , en } for V ∗ (the star used to denote the dual space is not to be confused with the HodgeP star!), with the inner product being expressed in terms of comn ponents as g = i,j=1 gij ei ⊗ ej , the ∗-operator is defined as the linear operator that maps the basis elements of Ωp (V ) as p |g| i1 l1 i1 ip ∗(e ∧ · · · ∧ e ) = g · · · g ip lp εl1 ···lp lp+1 ···ln elp+1 ∧ · · · ∧ eln . (n − p)! Here, |g| = det gij , and ε is the Levi-Civita permutation symbol This operator may be defined in a coordinate-free manner by the condition u ∧ ∗v = g(u, v) Vol(g) where the notation g(u, v) denotes the inner product on p-forms (in coordinates, g(u, v) = gi1 j1 · · · gip jp ui1 ...ip v j1 ...jp ) and Vol(g) is the unit volume form p associated to the metric. (in coordinates, Vol(g) = det(g)e1 ∧ · · · ∧ en ) Generally ∗∗ = (−1)p(n−p) id, where id is the identity operator in Ωp (V ). In three dimensions, ∗∗ = id for all p = 0, . . . , 3. On R3 with Cartesian coordinates, the metric tensor is g = dx ⊗ dx + dy ⊗ dy + dz ⊗ dz, and the Hodge star operator is ∗dx = dy ∧ dz, ∗dy = dz ∧ dx, ∗dz = dx ∧ dy. The Hodge star operation occurs most frequently in differential geometry in the case where M n is a n-dimensional orientable manifold with a Riemannian (or pseudo-Riemannian) tensor g and V is a cotangent vector space of M n . Also, one can extend this notion to antisymmetric tensor fields by computing Hodge star pointwise. ∗ hHodgeStarOperatori created: h2013-03-21i by: hrspuzioi version: h34120i Privacy setting: h1i hDefinitioni h53B21i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1