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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
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