In mathematics, the Hodge star operator or Hodge dual is an important linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finitedimensional oriented inner product space.
Contents

Dimensionalities and algebra 1

Extensions 2

Formal definition of the Hodge star of kvectors 3

Explanation 4

Computation of the Hodge star 5

Index notation for the star operator 6

Examples 7

Threedimensional example 7.1

Four dimensions 7.2

Inner product of kvectors 8

Duality 9

Hodge star on manifolds 10

Derivatives in three dimensions 11

Notes 12

References 13
Dimensionalities and algebra
Suppose that n is the dimensionality of the oriented inner product space and k is an integer such that 0 ≤ k ≤ n, then the Hodge star operator establishes a onetoone mapping from the space of kvectors to the space of (n − k)vectors. The image of a kvector under this mapping is called the Hodge dual of the kvector. The former space, of kvectors, has dimensionality

{n \choose k}
while the latter has dimensionality

{n \choose n  k},
and by the symmetry of the binomial coefficients, these two dimensionalities are equal. Two vector spaces over the same field with the same dimensionality are always isomorphic; but not necessarily in a natural or canonical way. The Hodge duality, however, in this case exploits the inner product and orientation of the vector space. It singles out a unique isomorphism, that reflects therefore the pattern of the binomial coefficients in algebra. This in turn induces an inner product on the space of kvectors. The 'natural' definition means that this duality relationship can play a geometrical role in theories.
The first interesting case is on threedimensional Euclidean space V. In this context the relevant row of Pascal's triangle reads

1, 3, 3, 1
and the Hodge dual sets up an isomorphism between the two threedimensional spaces, which are V itself and the space of wedge products of two vectors from V. See the Examples section for details. In this case the content is just that of the cross product of traditional vector calculus. While the properties of the cross product are special to three dimensions, the Hodge dual applies to all dimensionalities.
Extensions
Since the space of alternating linear forms in k arguments on a vector space is naturally isomorphic to the dual of the space of kvectors over that vector space, the Hodge dual can be defined for these spaces as well. As with most constructions from linear algebra, the Hodge dual can then be extended to a vector bundle. Thus a context in which the Hodge dual is very often seen is the exterior algebra of the cotangent bundle (i.e. the space of differential forms on a manifold) where it can be used to construct the codifferential from the exterior derivative, and thus the Laplacede Rham operator, which leads to the Hodge decomposition of differential forms in the case of compact Riemannian manifolds.
Formal definition of the Hodge star of kvectors
The Hodge star operator on a vector space V with a nondegenerate symmetric bilinear form (herein referred to as the inner product) is a linear operator on the exterior algebra of V, mapping kvectors to (n − k)vectors where n = dim V, for 0 ≤ k ≤ n. It has the following property, which defines it completely: given two kvectors α, β

\alpha \wedge (\star \beta) = \langle \alpha,\beta \rangle \omega
where \langle \cdot,\cdot \rangle denotes the inner product on kvectors and ω is the preferred unit nvector.
The inner product \langle \cdot,\cdot \rangle on kvectors is extended from that on V by requiring that

\langle \alpha,\beta \rangle = \det \left (\left \langle \alpha_i,\beta_j \right \rangle \right )
for any decomposable kvectors \alpha = \alpha_1 \wedge \dots \wedge \alpha_k and \beta = \beta_1 \wedge \dots \wedge \beta_k.
The unit nvector ω is unique up to a sign. The preferred choice of ω defines an orientation on V.
Explanation
Let W be a vector space, with an inner product \langle\cdot, \cdot\rangle_W. The Riesz representation theorem states that for every continuous (every in the finitedimensional case) linear functional f \in W^* there exists a unique vector v in W such that f(w) = \langle w, v \rangle_W for all w in W. The map W^* \to W given by f \mapsto v is an isomorphism. This holds for all vector spaces with an inner product, and can be used to explain the Hodge dual.
Let V be an ndimensional vector space with basis \{e_1,\ldots,e_n\}. For 0 ≤ k ≤ n, consider the exterior power spaces \bigwedge^k V and \bigwedge^{nk} V. For

\lambda \in \bigwedge^k V, \quad \theta \in \bigwedge^{nk} V,
we have

\lambda \wedge \theta \in \bigwedge^n V.
There is, up to a scalar, only one nvector, namely e_1\wedge\ldots\wedge e_n. In other words, \lambda \wedge \theta must be a scalar multiple of e_1\wedge\ldots\wedge e_n for all \lambda \in \bigwedge^k V and \theta \in \bigwedge^{nk} V.
Consider a fixed \lambda \in \bigwedge^k V. There exists a unique linear function

f_{\lambda} \in \left(\bigwedge^{nk} V\right)^{\! *}
such that

\forall \theta \in \bigwedge^{nk} V: \qquad \lambda \wedge \theta = f_{\lambda}(\theta) \, (e_1\wedge\ldots\wedge e_n).
This f_{\lambda}(\theta) is the scalar multiple mentioned in the previous paragraph. If \langle\cdot, \cdot\rangle denotes the inner product on (n − k)vectors, then there exists a unique (n − k)vector, say

\star \lambda \in \bigwedge^{nk} V,
such that

\forall \theta \in \bigwedge^{nk} V: \qquad f_{\lambda}(\theta) = \langle \theta, \star \lambda\rangle.
This (n − k)vector ★λ is the Hodge dual of λ, and is the image of the f_{\lambda} under the isomorphism induced by the inner product,

\left(\bigwedge^{nk} V\right)^{\! *} \cong \bigwedge^{nk} V.
Thus,

\star : \bigwedge^{k} V \to \bigwedge^{nk} V.
Computation of the Hodge star
Given an orthonormal basis (e_1,\cdots,e_n) ordered such that \omega = e_1\wedge \cdots \wedge e_n, we see that

\star (e_{i_1} \wedge e_{i_2}\wedge \cdots \wedge e_{i_k})= e_{i_{k+1}} \wedge e_{i_{k+2}} \wedge \cdots \wedge e_{i_n},
where (i_1, i_2, \cdots, i_n) is an even permutation of {1, 2, ..., n}.
Of these n! \over 2 relations, only n \choose k are independent. The first one in the usual lexicographical order reads

\star (e_1\wedge e_2\wedge \cdots \wedge e_k)= e_{k+1}\wedge e_{k+2}\wedge \cdots \wedge e_n.
Index notation for the star operator
Using index notation, the Hodge dual is obtained by contracting the indices of a kform with the ndimensional completely antisymmetric LeviCivita tensor. This differs from the LeviCivita symbol by a factor of det g^{1/2}, where g is the matrix of an inner product (the metric tensor) with respect to the basis. The absolute value of the determinant is necessary if g is not positivedefinite, e.g. for tangent spaces to Lorentzian manifolds.
Thus one writes^{[1]}

(\star \eta)_{i_1,i_2,\ldots,i_{nk}} = \frac{1}{(nk)!} \eta^{j_1,\ldots,j_k}\,\sqrt {\det g} \,\epsilon_{j_1,\ldots,j_k,i_1,\ldots,i_{nk}}
where η is an arbitrary antisymmetric tensor in k indices. It is understood that indices are raised and lowered using the same inner product g as in the definition of the LeviCivita tensor. Although one can take the star of any tensor, the result is antisymmetric, since the symmetric components of the tensor completely cancel out when contracted with the completely antisymmetric LeviCivita symbol.
Examples
A common example of the star operator is the case n = 3, when it can be taken as the correspondence between the vectors and the skewsymmetric matrices of that size. This is used implicitly in vector calculus, for example to create the cross product vector from the wedge product of two vectors. Specifically, for Euclidean R^{3}, one easily finds that

\star \mathrm{d}x=\mathrm{d}y\wedge \mathrm{d}z

\star \mathrm{d}y=\mathrm{d}z\wedge \mathrm{d}x

\star \mathrm{d}z=\mathrm{d}x\wedge \mathrm{d}y
where dx, dy and dz are the standard orthonormal differential oneforms on R^{3}. The Hodge dual in this case clearly relates the crossproduct to the wedge product in three dimensions. A detailed presentation not restricted to differential geometry is provided next.
Threedimensional example
Applied to three dimensions, the Hodge dual provides an isomorphism between axial vectors and bivectors, so each axial vector a is associated with a bivector A and vice versa, that is:^{[2]}

\mathbf{A} = \star \mathbf{a}\qquad\mathbf{a} = \star \mathbf{A}
where ★ indicates the dual operation. These dual relations can be implemented using multiplication by the unit pseudoscalar in Cℓ_{3}(R),^{[3]} i = e_{1}e_{2}e_{3} (the vectors {e_{ℓ}} are an orthonormal basis in three dimensional Euclidean space) according to the relations:^{[4]}

\mathbf{A} = \mathbf{a}i\,,\quad\mathbf{a} =  \mathbf{A} i.
The dual of a vector is obtained by multiplication by i, as established using the properties of the geometric product of the algebra as follows:

\begin{align} \mathbf{a}i &= \left(a_1 \mathbf{e_1} + a_2 \mathbf{e_2} +a_3 \mathbf {e_3}\right) \mathbf {e_1 e_2 e_3} \\ &= a_1 \mathbf{e_2 e_3} (\mathbf{e_1})^2 + a_2 \mathbf{e_3 e_1}(\mathbf{e_2})^2 +a_3 \mathbf{e_1 e_2}(\mathbf{e_3})^2 \\ &= a_1 \mathbf{e_2 e_3} +a_2 \mathbf{e_3 e_1} +a_3 \mathbf{e_1 e_2} \\ &= (\star \mathbf a ) \end{align}
and also, in the dual space spanned by {e_{ℓ}e_{m}}:

\begin{align} \mathbf{A} i &= \left(A_1 \mathbf{e_2e_3} + A_2 \mathbf{e_3e_1} +A_3 \mathbf {e_1e_2}\right) \mathbf {e_1 e_2 e_3} \\ &= A_1 \mathbf{e_1} (\mathbf{e_2 e_3})^2 +A_2 \mathbf{e_2} (\mathbf{e_3 e_1})^2 +A_3 \mathbf{e_3}(\mathbf{e_1 e_2})^2 \\ &=\left( A_1 \mathbf{e_1} + A_2 \mathbf{e_2} + A_3 \mathbf{e_3} \right) \\ &=  (\star \mathbf A ) \end{align}
In establishing these results, the identities are used:

(\mathbf{e_1e_2})^2 =\mathbf{e_1e_2e_1e_2}= \mathbf{e_1e_2e_2e_1} = 1
and:

\mathit{i}^2 =(\mathbf{e_1e_2e_3})^2 =\mathbf{e_1e_2e_3e_1e_2e_3}= \mathbf{e_1e_2e_3e_3e_1e_2} = \mathbf{e_1e_2e_1e_2} = 1.
These relations between the dual ★ and i apply to any vectors. Here they are applied to relate the axial vector created as the cross product a = u × v to the bivectorvalued exterior product A = u ∧ v of two polar (that is, not axial) vectors u and v; the two products can be written as determinants expressed in the same way:

\mathbf a = \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3\\u_1 & u_2 & u_3\\v_1 & v_2 & v_3 \end{vmatrix}\,,\quad\mathbf A = \mathbf{u} \wedge \mathbf{v} = \begin{vmatrix} \mathbf{e}_{23} & \mathbf{e}_{31} & \mathbf{e}_{12}\\u_1 & u_2 & u_3\\v_1 & v_2 & v_3 \end{vmatrix},
using the notation e_{ℓm} = e_{ℓ}e_{m}. These expressions show these two types of vector are Hodge duals:^{[2]}

\star (\mathbf u \wedge \mathbf v )=\mathbf {u \times v}\,,\quad\star (\mathbf u \times \mathbf v ) = \mathbf u \wedge \mathbf v,
as a result of the relations:

\star \mathbf e_{\ell} = \mathbf e_{\ell} \mathit i =\mathbf e_{\ell} \mathbf{e_1e_2e_3} = \mathbf e_m \mathbf e_n \,,
with ℓ, m, n cyclic,
and:

\star ( \mathbf e_{\ell} \mathbf e_m ) =( \mathbf e_{\ell} \mathbf e_m )\mathit{i} =\left( \mathbf e_{\ell} \mathbf e_m \right)\mathbf{e_1e_2e_3} =\mathbf e_{n}
also with ℓ, m, n cyclic.
Using the implementation of ★ based upon i, the commonly used relations are:^{[5]}

\mathbf {u \times v} = (\mathbf u \wedge \mathbf v ) i \,,\quad \mathbf u \wedge \mathbf v = (\mathbf {u \times v} ) i \ .
Four dimensions
In case n = 4, the Hodge dual acts as an endomorphism of the second exterior power (i.e. it maps twoforms to twoforms, since 4 − 2 = 2). It is an involution, so it splits it into selfdual and antiselfdual subspaces, on which it acts respectively as +1 and −1.
Another useful example is n = 4 Minkowski spacetime with metric signature (+ − − −) and coordinates (t, x, y, z) where (using \varepsilon_{0123} = 1)

\star \mathrm{d}t=\mathrm{d}x\wedge \mathrm{d}y \wedge\mathrm{d}z

\star \mathrm{d}x=\mathrm{d}t\wedge \mathrm{d}y \wedge\mathrm{d}z

\star \mathrm{d}y=\mathrm{d}t\wedge \mathrm{d}z \wedge\mathrm{d}x

\star \mathrm{d}z=\mathrm{d}t\wedge \mathrm{d}x \wedge\mathrm{d}y
for oneforms while

\star (\mathrm{d}t \wedge\mathrm{d}x) =  \mathrm{d}y\wedge \mathrm{d}z

\star (\mathrm{d}t \wedge\mathrm{d}y) = \mathrm{d}x\wedge \mathrm{d}z

\star (\mathrm{d}t \wedge\mathrm{d}z) =  \mathrm{d}x\wedge \mathrm{d}y

\star (\mathrm{d}x \wedge\mathrm{d}y) = \mathrm{d}t\wedge \mathrm{d}z

\star (\mathrm{d}x \wedge\mathrm{d}z) =  \mathrm{d}t\wedge \mathrm{d}y

\star (\mathrm{d}y \wedge\mathrm{d}z) = \mathrm{d}t\wedge \mathrm{d}x
for twoforms.
Inner product of kvectors
The Hodge dual induces an inner product on the space of kvectors, that is, on the exterior algebra of V. Given two kvectors η and ζ, one has

\zeta\wedge \star \eta = \langle\zeta, \eta \rangle\;\omega
where ω is the normalised nform (i.e. ω ∧ ★ω = ω). In the calculus of exterior differential forms on a pseudoRiemannian manifold of dimension n, the normalised nform is called the volume form and can be written as

\omega=\sqrt{ \left \det [g_{ij}] \right }\;\mathrm{d}x^1\wedge\cdots\wedge \mathrm{d}x^n
where \left[ g_{ij} \right] is the matrix of components of the metric tensor on the manifold in the coordinate basis.
If an inner product is given on \Lambda^k(V) , then this equation can be regarded as an alternative definition of the Hodge dual.^{[6]} The wedge products of elements of an orthonormal basis in V form an orthonormal basis of the exterior algebra of V.
Duality
The Hodge star defines a dual in that when it is applied twice, the result is an identity on the exterior algebra, up to sign. Given a kvector η in Λ^{k}(V) in an ndimensional space V, one has

\star {\star \eta}=(1)^{k(nk)}s\eta
where s is related to the signature of the inner product on V. Specifically, s is the sign of the determinant of the inner product tensor. Thus, for example, if n = 4 and the signature of the inner product is either (+ − − −) or (− + + +) then s = −1. For ordinary Euclidean spaces, the signature is always positive, and so s = 1. When the Hodge star is extended to pseudoRiemannian manifolds, then the above inner product is understood to be the metric in diagonal form.
Note that the above identity implies that the inverse of ★ can be given as

\begin{cases}\star^{1}:\Lambda^k \to \Lambda^{nk} \\ \eta \mapsto (1)^{k(nk)}s{\star \eta} \end{cases}
Note that if n is odd k(n − k) is even for any k whereas if n is even k(n − k) has the parity of k. Therefore:

\begin{cases} \star^{1} = s\star & n \text{ is odd} \\ \star^{1} = (1)^k s\star & n \text{ is even} \end{cases}
where k is the degree of the forms operated on.
Hodge star on manifolds
One can repeat the construction above for each cotangent space of an ndimensional oriented Riemannian or pseudoRiemannian manifold, and get the Hodge dual (n − k)form, of a kform. The Hodge star then induces an L^{2}norm inner product on the differential forms on the manifold. One writes

(\eta,\zeta)=\int_M \eta\wedge \star \zeta = \int_M \langle \eta, \zeta \rangle \; \mathrm{d} \text{Vol}
for the inner product of sections η and ζ of \Lambda^k(T^*M). (The set of sections is frequently denoted as \Omega^k(M)=\Gamma(\Lambda^k(T^*M)). Elements of \Omega^k(M) are called exterior kforms).
More generally, in the nonoriented case, one can define the hodge star of a kform as a (n − k)pseudo differential form; that is, a differential forms with values in the canonical line bundle.
The codifferential
The most important application of the Hodge dual on manifolds is to define the codifferential δ on kforms. Let

\delta = (1)^{n(k1) + 1}s\, {\star \mathrm{d}\star} = (1)^{k}\,{\star^{1}\mathrm{d}\star}
where d is the exterior derivative or differential, and s = 1 for Riemannian manifolds.

\mathrm{d}:\Omega^k(M)\to \Omega^{k+1}(M)
while

\delta:\Omega^k(M)\to \Omega^{k1}(M).
The codifferential is not an antiderivation on the exterior algebra, in contrast to the exterior derivative.
The codifferential is the adjoint of the exterior derivative, in that

(\eta,\delta \zeta) = (\mathrm{d}\eta,\zeta).
where ζ is a (k+1)form and η a kform. This identity follows from Stokes' theorem for smooth forms, when

\int_M \mathrm{d}(\eta \wedge \star \zeta)=0 =\int_M (\mathrm{d}\eta \wedge \star \zeta  \eta\wedge \star (1)^{k+1}\,{\star^{1}\mathrm{d}{\star \zeta}}) =(\mathrm{d}\eta,\zeta) (\eta,\delta\zeta)
i.e. when M has empty boundary or when η or ★ζ has zero boundary values (of course, true adjointness follows after continuous continuation to the appropriate topological vector spaces as closures of the spaces of smooth forms).
Notice that since the differential satisfies d^{2} = 0, the codifferential has the corresponding property

\! \delta^2 = s^2{\star \mathrm{d}{\star {\star \mathrm{d}{\star}}}} = (1)^{k(nk)} s^3{\star \mathrm{d}^2\star} = 0
The Laplace–deRham operator is given by

\! \Delta=(\delta+\mathrm{d})^2 = \delta \mathrm{d} + \mathrm{d}\delta
and lies at the heart of Hodge theory. It is symmetric:

(\Delta \zeta,\eta) = (\zeta,\Delta \eta)
and nonnegative:

(\Delta\eta,\eta) \ge 0.
The Hodge dual sends harmonic forms to harmonic forms. As a consequence of the Hodge theory, the de Rham cohomology is naturally isomorphic to the space of harmonic kforms, and so the Hodge star induces an isomorphism of cohomology groups

\star : H^k_\Delta(M)\to H^{nk}_\Delta(M),
which in turn gives canonical identifications via Poincaré duality of H^{ k}(M) with its dual space.
Derivatives in three dimensions
The combination of the ★ operator and the exterior derivative d generates the classical operators grad, curl, and div, in three dimensions. This works out as follows: d can take a 0form (function) to a 1form, a 1form to a 2form, and a 2form to a 3form (applied to a 3form it just gives zero). For a 0form, \omega=f(x,y,z), the first case written out in components is identifiable as the grad operator:

\mathrm{d}\omega=\frac{\partial f}{\partial x}\mathrm{d}x+\frac{\partial f}{\partial y}\mathrm{d}y+\frac{\partial f}{\partial z}\mathrm{d}z.
The second case followed by ★ is an operator on 1forms (\eta=A\,\mathrm{d}x+B\,\mathrm{d}y+C\,\mathrm{d}z) that in components is the curl operator:

\mathrm{d}\eta=\left({\partial C \over \partial y}  {\partial B \over \partial z}\right)\mathrm{d}y\wedge \mathrm{d}z + \left({\partial C \over \partial x}  {\partial A \over \partial z}\right)\mathrm{d}x\wedge \mathrm{d}z+\left({\partial B \over \partial x}  {\partial A \over \partial y}\right)\mathrm{d}x\wedge \mathrm{d}y.
Applying the Hodge star gives:

\star \mathrm{d}\eta=\left({\partial C \over \partial y}  {\partial B \over \partial z}\right)\mathrm{d}x  \left({\partial C \over \partial x}  {\partial A \over \partial z}\right)\mathrm{d}y+\left({\partial B \over \partial x}  {\partial A \over \partial y}\right)\mathrm{d}z.
The final case prefaced and followed by ★, takes a 1form (\eta=A\,\mathrm{d}x+B\,\mathrm{d}y+C\,\mathrm{d}z) to a 0form (function); written out in components it is the divergence operator:

\begin{align} \star\eta &= A\,\mathrm{d}y\wedge \mathrm{d}zB\,\mathrm{d}x\wedge \mathrm{d}z+C\,\mathrm{d}x\wedge \mathrm{d}y \\ \mathrm{d}{\star\eta} &= \left(\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}\right)\mathrm{d}x\wedge \mathrm{d}y\wedge \mathrm{d}z \\ \star \mathrm{d}{\star\eta} &= \frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}. \end{align}
One advantage of this expression is that the identity d^{2} = 0, which is true in all cases, sums up two others, namely that curl(grad( f )) = 0 and div(curl(F)) = 0. In particular, Maxwell's equations take on a particularly simple and elegant form, when expressed in terms of the exterior derivative and the Hodge star.
One can also obtain the Laplacian. Using the information above and the fact that Δ f = div grad f then for a 0form, \omega=f(x,y,z):

\Delta \omega =\star \mathrm{d}{\star \mathrm{d}\omega}= \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}
Notes

^ Frankel, T. (2012). The Geometry of Physics (3rd ed.). Cambridge University Press.

^ ^{a} ^{b} Pertti Lounesto (2001). "§3.6 The Hodge dual". Volume 286 of London Mathematical Society Lecture Note SeriesClifford Algebras and Spinors, (2nd ed.). Cambridge University Press. p. 39.

^ Venzo De Sabbata, Bidyut Kumar Datta (2007). "The pseudoscalar and imaginary unit". Geometric algebra and applications to physics. CRC Press. p. 53 ff.

^ William E Baylis (2004). "Chapter 4: Applications of Clifford algebras in physics". In Rafal Ablamowicz, Garret Sobczyk. Lectures on Clifford (geometric) algebras and applications. Birkhäuser. p. 100 ff.

^

^ Darling, R. W. R. (1994). Differential forms and connections. Cambridge University Press.
References

David Bleecker (1981) Gauge Theory and Variational Principles. AddisonWesley Publishing. ISBN 0201100967. Chpt. 0 contains a condensed review of nonRiemannian differential geometry.

Jurgen Jost (2002) Riemannian Geometry and Geometric Analysis. SpringerVerlag. ISBN 3540426272. A detailed exposition starting from basic principles; does not treat the pseudoRiemannian case.

Charles W. Misner, Kip S. Thorne, John Archibald Wheeler (1970) Gravitation. W.H. Freeman. ISBN 0716703440. A basic review of differential geometry in the special case of fourdimensional spacetime.

Steven Rosenberg (1997) The Laplacian on a Riemannian manifold. Cambridge University Press. ISBN 0521468310. An introduction to the heat equation and the AtiyahSinger theorem.

The Hodge Dual OperatorTevian Dray (1999) . A thorough overview of the definition and properties of the Hodge dual operator.
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