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{{Short description|involutive map on exterior algebras}}
In [[mathematics]], the '''Hodge star operator''' or '''Hodge star''' is a [[linear map]] defined on the [[exterior algebra]] of a finite-dimensional [[orientation (mathematics)|oriented]] [[vector space]] endowed with a [[Degenerate bilinear form|nondegenerate]] [[symmetric bilinear form]].  The result when applied to an element of the algebra is called the element's '''Hodge dual'''.  This map was introduced by [[W. V. D. Hodge]].

As an example, in an oriented 3-dimensional Euclidean space, every oriented plane has a unique corresponding unit [[normal vector]], and every vector corresponds to a unique oriented plane perpendicular to that vector. The Hodge star can be thought of as generalizing this relationship: in general, for an {{mvar|n}}-dimensional vector space, the Hodge star maps each {{mvar|k}}-vector to an {{nowrap|({{mvar|n}} − {{mvar|k}})}}-vector.

==Dimensions and algebra==
Suppose that {{mvar|n}} is the dimension of the oriented inner product space and {{mvar|k}} is an integer such that {{math|0 ≤ ''k'' ≤ ''n''}}, then the Hodge star operator establishes a one-to-one mapping from the space of [[p-vector|{{mvar|k}}-vectors]] to the space of {{math|(''n'' − ''k'')}}-vectors.  The image of a {{mvar|k}}-vector under this mapping is called its ''Hodge dual''. The former space, of {{mvar|k}}-vectors, has dimension <math>\tbinom{n}{k} </math>, while the latter has dimension <math>\tbinom{n}{n - k}</math>, which are equal by the symmetry of the [[binomial coefficient]]s. Equal-dimensional [[vector space]]s are always [[isomorphic]], but not necessarily in a [[Canonical map|canonical]] way. Given a nondegenerate symmetric bilinear form on the vector space, in this article referred to as the ''inner product'' (though it might not be positive-definite), Hodge duality exploits it and a choice of orientation to single out an isomorphism that parallels the combinatorial symmetry of binomial coefficients. This in turn induces an inner product on the space of {{mvar|k}}-vectors. The naturalness of this definition means the duality can play a role in differential geometry.

The first interesting case is on three-dimensional [[Euclidean space]] {{mvar|V}}. In this context the relevant row of [[Pascal's triangle]] reads

:{{math|1, 3, 3, 1}}

and the Hodge star sets up an isomorphism between the two three-dimensional spaces, which are {{mvar|V}} and the image space of the [[exterior product]] acting on pairs of vectors in {{mvar|V}}. See ''{{section link||Examples}}'' for details. In this case the Hodge star allows the definition of the [[cross product]] of traditional [[vector calculus]] in terms of the exterior product. While the properties of the cross product are specific to three dimensions, the Hodge star applies in an arbitrary number of dimensions.

==Extensions==
Since the space of alternating linear forms in {{mvar|k}} arguments on a vector space is canonically isomorphic to the dual of the space of {{mvar|k}}-vectors over that vector space, the Hodge star can be defined for these spaces as well.  As with most constructions from linear algebra, the Hodge star can then be extended to a [[vector bundle]].  Thus a context in which the Hodge star is often seen is the exterior algebra of the cotangent bundle, the space of differential forms on a manifold, where it can be used to construct the [[#Codifferential|codifferential]] from the [[exterior derivative]], and thus the [[Laplace–de Rham operator]], which leads to the [[Hodge decomposition]] of [[differential forms]] in the case of [[Compact space|compact]] [[Riemannian manifold]]s.

==Formal definition for {{math|''k''}}-vectors==
The '''Hodge star operator''' on a [[vector space]] {{math|''V''}} with an inner product is a linear operator on the [[exterior algebra]] of {{math|''V''}}, mapping {{mvar|k}}-vectors to {{math|(''n'' − ''k'')}}-vectors where {{math|''n'' {{=}} dim ''V''}}, for {{math|0 ≤ ''k'' ≤ ''n''}}. It has the following property, which defines it completely: given two {{math|''k''}}-vectors {{math|''α'', ''β''}},

:<math>\alpha \wedge (\star \beta) = \langle \alpha,\beta \rangle \,\omega</math>

where <math>\langle \cdot,\cdot \rangle</math> denotes the [[Exterior algebra#Inner product|inner product on {{mvar|k}}-vectors]] and {{mvar|ω}} is the preferred unit {{mvar|n}}-vector.<ref name=HF>[[Harley Flanders]] (1963) ''Differential Forms with Applications to the Physical Sciences'', [[Academic Press]]</ref>{{rp|15}}

The inner product <math>\langle \cdot,\cdot \rangle</math> on {{mvar|k}}-vectors is extended from that on {{mvar|V}} by requiring that for any decomposable {{mvar|k}}-vectors <math>\alpha = \alpha_1 \wedge \cdots \wedge \alpha_k</math> and <math>\beta = \beta_1 \wedge \cdots \wedge \beta_k</math> it equals the [[determinant]]<ref name=HF/>{{rp|14}}

:<math> \langle \alpha,\beta \rangle = \det \left[ \left\langle \alpha_i,\beta_j \right\rangle \right] ,</math> sometimes called the [[Gram determinant]].

The unit {{mvar|n}}-vector {{mvar|ω}} is unique up to a sign. The preferred choice of {{mvar|ω}} defines an [[orientation (vector space)|orientation]] on {{mvar|V}}.

==Explanation==
Let {{mvar|W}} be a vector space, with an inner product <math>\langle\cdot, \cdot\rangle_W</math>. The [[Riesz representation theorem]] states that for every continuous (even in the infinite-dimensional case) [[linear functional]] <math>f \in W^*</math> there exists a unique vector {{mvar|v}} in {{mvar|W}} such that <math>f(w) = \langle w, v \rangle_W</math> for all {{mvar|w}} in {{mvar|W}}. The map <math>W^* \to W</math> given by <math>f \mapsto v</math> is an isomorphism. (If {{mvar|W}} is complex, the map is ''conjugate linear'' as opposed to complex linear.) This holds for ''all'' vector spaces with an inner product, and can be used to explain the Hodge star.

Let {{mvar|V}} be an {{mvar|n}}-dimensional vector space with basis <math>\{e_1,\ldots,e_n\}</math>. For {{math|0 ≤ ''k'' ≤ ''n''}}, consider the exterior power spaces <math>{\textstyle\bigwedge}^k V</math> and <math>{\textstyle\bigwedge}^{n-k} V</math>. For

:<math>\lambda \in {\textstyle\bigwedge}^k V, \quad \theta \in {\textstyle\bigwedge}^{n-k} V,</math>

we have

:<math>\lambda \wedge \theta \in {\textstyle\bigwedge}^n V.</math>

There is, up to a scalar, only one {{mvar|n}}-vector (an {{mvar|n}}-form), namely <math>e_1\wedge\ldots\wedge e_n</math>. In other words, <math>\lambda \wedge \theta</math> must be a scalar multiple of <math>e_1\wedge\ldots\wedge e_n</math> for all <math>\lambda \in {\textstyle\bigwedge}^k V</math> and <math>\theta \in {\textstyle\bigwedge}^{n-k} V</math>.

Consider a ''fixed'' <math>\lambda \in {\textstyle\bigwedge}^k V</math>. There exists a unique linear function

:<math>f_{\lambda} \in \left({\textstyle\bigwedge}^{n-k} V\right)^{\! *}</math>

such that

:<math>\forall \theta \in {\textstyle\bigwedge}^{n-k} V: \qquad \lambda \wedge \theta = f_{\lambda}(\theta) \, (e_1\wedge\ldots\wedge e_n).</math>

This <math>f_{\lambda}(\theta)</math> is the scalar multiple mentioned in the previous paragraph. If <math>\langle\cdot, \cdot\rangle</math> denotes the [[Exterior algebra#Inner product|inner product on {{math|(''n'' − ''k'')}}-vectors]], then there exists a unique {{math|(''n'' − ''k'')}}-vector, say

:<math>\star \lambda \in {\textstyle\bigwedge}^{n-k} V,</math>

such that

:<math>\forall \theta \in {\textstyle\bigwedge}^{n-k} V: \qquad f_{\lambda}(\theta) = \langle \theta, \star \lambda\rangle.</math>

This {{math|(''n'' − ''k'')}}-vector <math>\star\lambda</math> is the Hodge dual of {{mvar|λ}}, and is the image of the <math>f_{\lambda}</math> under the isomorphism induced by the inner product,

:<math>\left({\textstyle\bigwedge}^{n-k} V\right)^{\! *} \cong {\textstyle\bigwedge}^{n-k} V.</math>

Thus,

:<math> \star : {\textstyle\bigwedge}^{k} V \to {\textstyle\bigwedge}^{n-k} V.</math>

==Computation of the Hodge star==
Given an orthonormal basis <math>(e_1,\cdots,e_n)</math> ordered such that <math>\omega = e_1\wedge \cdots \wedge e_n</math>, for a positive-definite metric, we see that

:<math>\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},</math>

where <math>(i_1, i_2, \cdots, i_n)</math> is an even permutation of {{math|{1, 2, ..., ''n''}.}}

Of these <math>n! \over 2</math> relations, only <math>n \choose k</math> are independent. The first one in the usual [[lexicographical order]] reads

:<math>\star (e_1\wedge e_2\wedge \cdots \wedge e_k)= e_{k+1}\wedge e_{k+2}\wedge \cdots \wedge e_n.</math>

==Expression in index notation==
Using [[tensor index notation]], the Hodge dual of an arbitrary wedge product of one-forms is given by the following:

:<math>
\star\left(dx^{i_1} \wedge \dots \wedge dx^{i_k}\right) = \frac{\sqrt{|\det [g_{ab}]|}}{(n-k)!} g^{i_1 j_1} \cdots g^{i_k j_k} \varepsilon_{j_1 \dots j_n} dx^{j_{k+1}} \wedge \dots \wedge dx^{j_n}.
</math>

The symbol <math>\varepsilon_{j_1 \dots j_n}</math> is the [[Levi-Civita symbol]] defined so that <math>\varepsilon_{1 \dots n} = 1</math>, and <math>g_{ab}</math> is the metric and <math>g^{ij}</math> is its inverse. The factorial <math>(n-k)!</math> had to be inserted to account for double counting, and is not present if the summation indices are restricted so that <math>j_{k+1} < \dots < j_n</math>.  The absolute value of the determinant is necessary since it may be negative, e.g. for tangent spaces to [[Pseudo-Riemannian manifold#Lorentzian manifold|Lorentzian manifold]]s.

An arbitrary differential form can be expanded into its components as follows:

:<math>
\alpha = \frac{1}{k!}\alpha_{i_1, \dots, i_k} dx^{i_1}\wedge \dots \wedge dx^{i_k} = \sum_{i_1 < \dots < i_k} \alpha_{i_1, \dots, i_k} dx^{i_1}\wedge \dots \wedge dx^{i_k}
</math>

The factorial <math>k!</math> is again included to account for double counting. We would like to define the dual of the component <math>\alpha_{i_1, \dots, i_k}</math> so that the Hodge dual of the form is given by

:<math>
(\star\alpha) = \frac{1}{(n-k)!}(\star \alpha)_{i_{k+1}, \dots, i_n} dx^{i_{k+1}} \wedge \dots \wedge dx^{i_n}.
</math>

Thus using the expression for the Hodge dual of <math>dx^{i_1} \wedge \dots \wedge dx^{i_k}</math> given at the beginning of this section we find<ref>{{cite book |title=The Geometry of Physics |edition=3rd |first=T. |last=Frankel |publisher=Cambridge University Press |year=2012 |isbn=978-1-107-60260-1 }}</ref>

:<math>
(\star \alpha)_{i_{k+1}, \dots, i_n} = \frac{\sqrt{|\det [g_{ab}]|}}{k!} \alpha^{i_1, \dots, i_k}\, \,\varepsilon_{i_1, \dots, i_n}
</math>

Although one can apply this expression to any tensor <math>\alpha</math>, the result is antisymmetric, since contraction with the completely anti-symmetric Levi-Civita symbol cancels all but the totally antisymmetric part of the tensor.  It is thus equivalent to antisymmetrization followed by applying the Hodge star.

==Examples==

===Two dimensions===
In two dimensions with the normalized Euclidean metric and orientation given by the ordering {{math|(''x'', ''y'')}}, the Hodge star on {{math|''k''}}-forms is given by

:<math> {\star} \, 1 = dx \wedge dy </math>
:<math> {\star} \, dx = dy </math>
:<math> {\star} \, dy = -dx </math>
:<math> {\star} ( dx \wedge dy ) = 1 .</math>

On the complex plane regarded as a real vector space with the standard [[sesquilinear form]] as the metric, the Hodge star has the remarkable property that it is invariant under [[holomorphic]] changes of coordinate.
If {{math|''z'' {{=}} ''x'' + ''iy''}} is a holomorphic function of {{math|''w'' {{=}} ''u'' + ''iv''}}, then by the [[Cauchy–Riemann equations]] we have that {{math|{{sfrac|∂''x''|∂''u''}} {{=}} {{sfrac|∂''y''|∂''v''}}}} and {{math|{{sfrac|∂''y''|∂''u''}} {{=}} –{{sfrac|∂''x''|∂''v''}}}}. In the new coordinates

:<math> \alpha = p \,dx + q \,dy = \left( p \frac{\partial x}{\partial u} + q \frac{\partial y}{\partial u} \right) \,du + \left( p \frac{\partial x}{\partial v} + q \frac{\partial y}{\partial v} \right) \,dv = p_1 du + q_1 \, dv ,</math>

so that

: <math>
\begin{align}
  {\star}\alpha = -q_1 \,du + p_1 \,dv &= - \left( p \frac{\partial x}{\partial v} + q \frac{\partial y}{\partial v} \right) du + \left(p \frac{\partial x}{\partial u} + q \frac{\partial y}{\partial u} \right) dv \\[4pt]
                       &= -q \left( \frac{\partial x}{\partial u} du + \frac{\partial x}{\partial v} dv \right) + p \left( \frac{\partial y}{\partial u} du + \frac{\partial y}{\partial v} dv \right) \\[4pt]
& = -q\,dx + p\, dy,
\end{align}
</math>

proving the claimed invariance.

===Three dimensions===
A common example of the Hodge star operator is the case {{math|''n'' {{=}} 3}}, when it can be taken as the correspondence between the vectors and the 3×3 [[skew-symmetric matrix|skew-symmetric matrices]]. This is used implicitly in [[vector calculus]], for example to create the [[cross product]] vector from the [[Exterior algebra|wedge product]]  of two vectors. Specifically, for [[Euclidean space|Euclidean]] '''R'''<sup>3</sup>, one finds that

:<math>\begin{align}
  \star \,dx &= dy \wedge dz \\
  \star \,dy &= dz \wedge dx \\
  \star \,dz &= dx \wedge dy
\end{align}</math>

where {{math|''dx''}}, {{math|''dy''}} and {{math|''dz''}} are the standard orthonormal differential [[one-form]]s on '''R'''<sup>3</sup>. The Hodge star in this case clearly relates the cross-product to the wedge product in three dimensions. A detailed presentation not restricted to differential geometry is provided next.

Applied to three dimensions, the Hodge star provides an [[isomorphism]] between [[axial vector]]s and [[bivector]]s, so each axial vector '''a''' is associated with a bivector {{math|'''A'''}} and vice versa, that is:<ref name=Lounesto>{{cite book |title=Clifford Algebras and Spinors, ''Volume 286 of London Mathematical Society Lecture Note Series'' |author=Pertti Lounesto |url=https://books.google.com/books?id=E_xvJuA4M7QC&pg=PA39 |page=39 |chapter=§3.6 The Hodge dual |isbn=0-521-00551-5 |year=2001 |edition=2nd |publisher=Cambridge University Press}}</ref>

: <math>\mathbf{A} = \star \mathbf{a}\qquad\mathbf{a} = \star \mathbf{A}</math>

where <math>\star</math> is the Hodge star. These dual relations can be implemented using multiplication by the [[Pseudoscalar (Clifford algebra)#Unit pseudoscalar|unit pseudoscalar]] in [[Clifford algebra#Examples: real and complex Clifford algebras|Cl<sub>3</sub>('''R''')]],<ref name=Datta>{{cite book |title=Geometric algebra and applications to physics |chapter=The pseudoscalar and imaginary unit | url=https://books.google.com/books?id=AXTQXnws8E8C&pg=PA53 |page=53 ''ff'' |author=Venzo De Sabbata, Bidyut Kumar Datta |isbn=1-58488-772-9 | publisher=CRC Press |year=2007}}</ref> {{math|''i'' {{=}} '''e'''<sub>1</sub>'''e'''<sub>2</sub>'''e'''<sub>3</sub>}} (the vectors {{math|{'''e'''<sub>ℓ</sub>} }} are an orthonormal basis in three-dimensional Euclidean space) according to the relations<ref name=Baylis>{{cite book |title=Lectures on Clifford (geometric) algebras and applications |editor=Rafal Ablamowicz, Garret Sobczyk |page=100 ''ff'' |chapter=Chapter 4: Applications of Clifford algebras in physics |author=William E Baylis  |isbn=0-8176-3257-3 |year=2004 | publisher=Birkhäuser | url=https://books.google.com/books?id=oaoLbMS3ErwC&pg=PA100}}</ref>

: <math>\mathbf{A} = \mathbf{a}i\,,\quad\mathbf{a} = - \mathbf{A} i. </math>

The dual of a vector is obtained by multiplication by {{mvar|i}}, as established using the properties of the [[Geometric product#The geometric product|geometric product]] of the algebra as follows:

:<math>\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 \mathbf{e}_2 \mathbf{e}_3 \\
              &= a_1 \mathbf{e}_2 \mathbf{e}_3 (\mathbf{e}_1)^2 + a_2 \mathbf{e}_3 \mathbf{e}_1(\mathbf{e}_2)^2 + a_3  \mathbf{e}_1 \mathbf{e}_2(\mathbf{e}_3)^2 \\
              &= a_1 \mathbf{e}_2 \mathbf{e}_3 + a_2 \mathbf{e}_3 \mathbf{e}_1 + a_3 \mathbf{e}_1 \mathbf{e}_2 \\
              &= (\star \mathbf{a})
\end{align}</math>

and also, in the dual space spanned by {{math|{'''e'''<sub>ℓ</sub>'''e'''<sub>''m''</sub>}:}}

:<math>\begin{align}
  \mathbf{A} i &= \left(A_1 \mathbf{e}_2\mathbf{e}_3 + A_2 \mathbf{e}_3\mathbf{e}_1 + A_3 \mathbf{e}_1\mathbf{e}_2\right) \mathbf{e}_1 \mathbf{e}_2 \mathbf{e}_3 \\
               &= A_1 \mathbf{e}_1 (\mathbf{e}_2 \mathbf{e}_3)^2 + A_2 \mathbf{e}_2 (\mathbf{e}_3 \mathbf{e}_1)^2 + A_3 \mathbf{e}_3(\mathbf{e}_1 \mathbf{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}</math>

In establishing these results, the identities are used:

:<math>(\mathbf{e}_1\mathbf{e}_2)^2 = \mathbf{e}_1\mathbf{e}_2\mathbf{e}_1\mathbf{e}_2= -\mathbf{e}_1\mathbf{e}_2\mathbf{e}_2\mathbf{e}_1 = -1</math>

and:

:<math>\mathit{i}^2 = (\mathbf{e}_1\mathbf{e}_2\mathbf{e}_3)^2 = \mathbf{e}_1\mathbf{e}_2\mathbf{e}_3\mathbf{e}_1\mathbf{e}_2\mathbf{e}_3 = \mathbf{e}_1\mathbf{e}_2\mathbf{e}_3\mathbf{e}_3\mathbf{e}_1\mathbf{e}_2 = \mathbf{e}_1\mathbf{e}_2\mathbf{e}_1\mathbf{e}_2 = -1.</math>

These relations between the dual <math>\star</math> and {{mvar|i}} apply to any vectors. Here they are applied to relate the axial vector created as the [[cross product]] {{math|1='''a''' = '''u''' × '''v'''}} to the bivector-valued [[exterior product]] {{math|1='''A''' = '''u''' ∧ '''v'''}} of two [[polar vectors|polar]] (that is, not axial) vectors {{math|'''u'''}} and {{math|'''v'''}}; the two products can be written as [[determinant]]s expressed in the same way:

: <math>\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}_{2}\mathbf{e}_{3} & \mathbf{e}_{3}\mathbf{e}_{1} & \mathbf{e}_{1}\mathbf{e}_{2}\\u_1 & u_2 & u_3\\v_1 & v_2 & v_3 \end{vmatrix}.</math>

These expressions show these two types of vector are Hodge duals:<ref name=Lounesto/>

:<math>\star (\mathbf{u} \wedge \mathbf{v}) = \mathbf{u \times v}\,,\quad\star (\mathbf{u} \times \mathbf {v}) = \mathbf{u} \wedge \mathbf{v},</math>

as a result of the relations:

:<math>\star \mathbf{e}_\ell = \mathbf{e}_\ell \mathit{i} = \mathbf{e}_\ell \mathbf{e}_1\mathbf{e}_2\mathbf{e}_3 = \mathbf{e}_m \mathbf{e}_n \,, </math>

with {{math|ℓ, ''m'', ''n''}} cyclic,

and:

:<math>\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}_1\mathbf{e}_2\mathbf{e}_3 = \mathbf{e}_n </math>

also with {{math|''ℓ'', ''m'', ''n''}} cyclic.

Using the implementation of <math>\star</math> based upon {{mvar|i}}, the commonly used relations are:<ref name=Hestenes>{{cite book |title=New foundations for classical mechanics: Fundamental Theories of Physics  |isbn=0-7923-5302-1 |edition=2nd |year=1999 |publisher=Springer |chapter=The vector cross product |authorlink = David Hestenes|author=David Hestenes |url=https://books.google.com/books?id=AlvTCEzSI5wC&pg=PA60 |page=60 }}</ref>

:<math> \mathbf{u \times v} = -(\mathbf{u} \wedge \mathbf{v}) i  \,,\quad  \mathbf{u} \wedge \mathbf{v} = (\mathbf{u \times v} )  i \ . </math>

===Four dimensions===
In case {{math|1=''n'' = 4}}, the Hodge star acts as an [[endomorphism]] of the second exterior power (i.e. it maps two-forms to two-forms, since {{math|1=4 − 2 = 2}}). If the signature of the [[metric tensor]] is all positive, i.e. on a [[Riemannian manifold]], then the Hodge star is an [[involution (mathematics)|involution]]; if the signature is mixed, then application twice will return the argument up to a sign – see ''{{section link|#Duality}}'' below. For example, in Minkowski spacetime where {{math|''n'' {{=}} 4}} with metric signature {{math|(+ − − −)}} and coordinates {{math|(''t'', ''x'', ''y'', ''z'')}} where (using <math>\varepsilon_{0123} = 1</math>):

:<math>\begin{align}
  \star dt &=  dx \wedge dy \wedge dz \\
  \star dx &=  dt \wedge dy \wedge dz \\
  \star dy &= -dt \wedge dx \wedge dz \\
  \star dz &=  dt \wedge dx \wedge dy
\end{align}</math>

for [[one-form]]s while

:<math>\begin{align}
  \star (dt \wedge dx) &= - dy \wedge dz \\
  \star (dt \wedge dy) &=   dx \wedge dz \\
  \star (dt \wedge dz) &= - dx \wedge dy \\
  \star (dx \wedge dy) &=   dt \wedge dz \\
  \star (dx \wedge dz) &= - dt \wedge dy \\
  \star (dy \wedge dz) &=   dt \wedge dx
\end{align}</math>

for [[two-form]]s. Because their determinants are the same in both {{math|(+ − − −)}} and {{math|(− + + +)}}, the signs of the Minkowski space two-form duals depend only on the chosen orientation.{{verify source|date=October 2017}}

An easy rule to remember for the above Hodge operations is that given a form <math>\alpha</math>, its Hodge dual <math>{\star}\alpha</math> may be obtained by writing the components not involved in <math>\alpha</math> in an order such that <math>\alpha \wedge (\star \alpha) = dt \wedge dx \wedge dy \wedge dz </math>.{{verify source|date=September 2019}} An extra minus sign will enter only if <math>\alpha</math> does not contain <math>dt</math>. (The latter convention stems from the choice {{math|(+ − − −)}} for the metric signature. For {{math|(− + + +)}}, one puts in a minus sign only if <math>\alpha</math> involves <math>dt</math>.)

==Inner product of {{math|''k''}}-vectors==
The Hodge star induces an inner product on the space of {{math|''k''}}-vectors, that is, on the [[exterior algebra]] of {{math|''V''}}.  Given two {{math|''k''}}-vectors {{math|''η''}} and {{math|''ζ''}}, one has

:<math>\zeta\wedge \star \eta = \langle\zeta, \eta \rangle\,\omega ,</math>

where <math>\omega = \star 1</math>. In the calculus of exterior [[differential form]]s on a [[pseudo-Riemannian manifold]] of dimension {{math|''n''}}, the normalised {{math|''n''}}-form is called the [[volume form]] and can be written as

:<math>\omega = \sqrt{ \left| \det [g_{ij}] \right| }\;dx^1\wedge\cdots\wedge dx^n ,</math>

where <math> \left[ g_{ij} \right] </math> is the matrix of components of the [[metric tensor]] on the manifold in the [[coordinate basis]].

If an inner product is given on <math> {\textstyle\bigwedge}^k(V) </math>, then this equation can be regarded as an alternative definition of the Hodge star.<ref>{{cite book | last = Darling| first = R. W. R. | title = Differential forms and connections| publisher = Cambridge University Press| year = 1994 }}</ref>

The ordered wedge products of {{math|''k''}} distinct orthonormal basis vectors of {{math|''V''}} form an ''orthonormal'' basis on each subspace <math>{\textstyle\bigwedge}^k(V)</math> of the exterior algebra of {{mvar|V}}.

==Duality==
Applying the Hodge star twice leaves a {{math|''k''}}-vector unchanged, up to sign. Given a {{mvar|k}}-vector {{math|''η''}} in {{math|⋀<sup>''k''</sup>(''V'')}} in an {{mvar|n}}-dimensional space {{mvar|V}}, one has

:<math>{\star} {\star} \eta = (-1)^{k(n-k)} s \eta ,</math>

where {{mvar|s}} is related to the [[metric signature|signature]] of the inner product on {{mvar|V}}.  Specifically, {{mvar|s}} is the sign of the [[determinant]] of the matrix representation of the inner product tensor with respect to any basis. Thus, for example, if {{math|''n'' {{=}} 4}} and the signature of the inner product is either {{math|(+ − − −)}} or {{math|(− + + +)}} then {{math|''s'' {{=}} −1}}. For Riemannian manifolds (including Euclidean spaces), the signature is always positive, and so {{math|''s'' {{=}} 1}}.

The above identity implies that the inverse of <math>\star</math> can be given as

:<math> \begin{align} {\star}^{-1}: ~& \Lambda^k \to \Lambda^{n-k} \\ & \eta \mapsto (-1)^{k(n-k)} s {\star} \eta \end{align}</math>

If {{mvar|n}} is odd then {{math|''k''(''n'' − ''k'')}} is even for any {{mvar|k}}, whereas if {{mvar|n}} is even then {{math|''k''(''n'' − ''k'')}} has the parity of {{mvar|k}}. Therefore:

:<math>{\star}^{-1} = \begin{cases} s {\star} & n \text{ is odd} \\ (-1)^k s {\star} & n \text{ is even} \end{cases}</math>

where {{mvar|k}} is the degree of the element operated on.

==On manifolds==
Applying the construction above to each [[cotangent space]] of an {{mvar|n}}-dimensional oriented [[Riemannian manifold|Riemannian]] or [[pseudo-Riemannian manifold]], we can obtain an object known as the '''Hodge dual''' of a [[differential form|{{mvar|k}}-form]].  To be explicit –

For any {{math|''k''}}-form {{math|''ζ''}} we define <math>{\star} \zeta</math> as the unique {{math|(''n'' − ''k'')}}-form satisfying

:<math>\eta\wedge {\star} \zeta = \langle \eta, \zeta \rangle \; \omega </math>

for every {{math|''k''}}-form {{mvar|η}} (here the inner product on forms and the volume form {{math|''ω''}} are induced by the Riemannian metric tensor in the usual way; greater understanding of these objects can be found by learning about the [[Exterior algebra#Inner product|inner product on {{mvar|k}}-forms]] and the [[Volume form#Riemannian volume form|volume form]]).

The Hodge star is thus related to the {{math|''L''<sup>2</sup>}} inner product on {{math|''k''}}-forms by the formula:

:<math>(\eta,\zeta)=\int_M \eta\wedge {\star} \zeta.</math>

for {{math|''k''}}-forms  {{mvar|η}} and {{mvar|ζ}}. (We can also see this as an inner product on  [[fibre bundle|sections]]  of <math>{\textstyle\bigwedge}^k(\text{T}^*M)</math>.  The set of sections is frequently denoted as <math>\Omega^k(M) = \Gamma\left({\textstyle\bigwedge}^k\left(\text{T}^*M\right)\right)</math>.  Each element of <math>\Omega^k(M)</math> is a {{math|''k''}}-form.)

More generally, in the non-oriented case, one can define the Hodge star of a {{mvar|k}}-form as a {{math|(''n'' − ''k'')}}-[[pseudotensor|pseudo differential form]]; that is, a differential form with values in the [[Canonical bundle|canonical line bundle]].

===Codifferential<!-- This section is linked from [[Differential form]] -->===
The most important application of the Hodge star on manifolds is to define the '''codifferential''' {{mvar|δ}} on {{mvar|k}}-forms.  Let

:<math>\delta = (-1)^{n(k-1) + 1} s\ {\star} d {\star} = (-1)^{k}\, {\star}^{-1} d {\star} </math>

where {{math|''d''}} is the [[exterior derivative]] or differential, and {{math|''s'' {{=}} 1}} for Riemannian manifolds.

:<math>d:\Omega^k(M)\to \Omega^{k+1}(M)</math>

while

:<math>\delta:\Omega^k(M)\to \Omega^{k-1}(M).</math>

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

:<math> (\eta,\delta \zeta) = (d\eta,\zeta). </math>

where {{mvar|ζ}} is a {{math|(''k'' + 1)}}-form and {{mvar|η}} a {{mvar|k}}-form. This identity follows from Stokes' theorem for smooth forms, when

:<math>
\int_M d (\eta \wedge {\star} \zeta) = 0 =
\int_M \left(d \eta \wedge {\star} \zeta - \eta \wedge {\star} (-1)^{k+1}\,{\star}^{-1} d {\star} \zeta\right) =
(d\eta,\zeta) - (\eta,\delta\zeta)
</math>

i.e. when {{mvar|M}} has empty boundary or when {{mvar|η}} or <math>\star\zeta</math> 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).

Since the differential satisfies {{math|''d''{{i sup|2}} {{=}} 0}}, the codifferential has the corresponding property

:<math>\delta^2 = s^2 {\star} d {\star} {\star} d {\star} = (-1)^{k(n-k)} s^3 {\star} d^2 {\star} = 0 </math>

The [[Laplace–Beltrami operator|Laplace–deRham]] operator is given by

:<math>\Delta = (\delta + d)^2 = \delta d + d\delta</math>

and lies at the heart of [[Hodge theory]]. It is symmetric:

:<math>(\Delta \zeta,\eta) = (\zeta,\Delta \eta)</math>

and non-negative:

:<math>(\Delta\eta,\eta) \ge 0.</math>

The Hodge star 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 {{mvar|k}}-forms, and so the Hodge star induces an isomorphism of cohomology groups

:<math>{\star} : H^k_\Delta (M) \to H^{n-k}_\Delta(M),</math>

which in turn gives canonical identifications via [[Poincaré duality]] of {{math|''H<sup> k</sup>''(''M'')}} with its [[dual space]].

==Derivatives in three dimensions==
The combination of the <math>\star</math> operator and the [[exterior derivative]] {{math|''d''}} generates the classical operators [[gradient|{{math|grad}}]], [[Curl (mathematics)|{{math|curl}}]], and [[divergence|{{math|div}}]], in three-dimensional Euclidean space. This works out as follows: {{math|''d''}} can take a 0-form (function) to a 1-form, a 1-form to a 2-form, and a 2-form to a 3-form (applied to a 3-form it just gives zero). For a 0-form, <math>\omega=f(x,y,z)</math>, the first case written out in components is identifiable as the grad operator:

:<math>d\omega=\frac{\partial f}{\partial x} \, dx+\frac{\partial f}{\partial y} \, dy + \frac{\partial f}{\partial z} \, dz.</math>

The second case followed by <math>\star</math> is an operator on 1-forms (<math>\eta=A\,dx+B\,dy+C\,dz</math>) that in components is the curl operator:

:<math>d\eta=\left({\partial C \over \partial y} - {\partial B \over \partial z}\right)dy\wedge dz  + \left({\partial C \over \partial x} - {\partial A \over \partial z}\right)dx\wedge dz+\left({\partial B \over \partial x} - {\partial A \over \partial y}\right)dx\wedge dy.</math>

Applying the Hodge star gives:

:<math>\star d\eta=\left({\partial C \over \partial y} - {\partial B \over \partial z} \right) \, dx  - \left({\partial C \over \partial x} - {\partial A \over \partial z} \right) \, dy+\left({\partial B \over \partial x} - {\partial A \over \partial y}\right) \, dz.</math>

The final case prefaced and followed by <math>\star</math>, takes a 1-form (<math>\eta=A\,dx+B\,dy+C\,dz</math>) to a 0-form (function); written out in components it is the divergence operator:

:<math>\begin{align}
\star\eta &= A\,dy\wedge dz-B\,dx\wedge dz+C\,dx\wedge dy \\
d{\star\eta} &= \left(\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}\right)dx\wedge dy\wedge dz \\
\star d{\star\eta} &= \frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}.
\end{align}</math>

One advantage of this expression is that the identity {{math|''d''{{i sup|2}} {{=}} 0}}, which is true in all cases, sums up two others, namely that {{math|curl grad ''f'' {{=}} 0}} and {{math|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 {{math|Δ ''f'' {{=}} div grad ''f''}} then for a 0-form, <math>\omega=f(x,y,z)</math>:

:<math> \Delta \omega =\star d{\star d\omega}= \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}</math>

==Notes==
{{Reflist}}

==References==
* David Bleecker (1981) ''Gauge Theory and Variational Principles''. Addison-Wesley Publishing. {{isbn|0-201-10096-7}}. Chpt. 0 contains a condensed review of non-Riemannian differential geometry.
* Jurgen Jost (2002) ''Riemannian Geometry and Geometric Analysis''. Springer-Verlag. {{isbn|3-540-42627-2}}. A detailed exposition starting from basic principles; does not treat the pseudo-Riemannian case.
* [[Charles W. Misner]], [[Kip S. Thorne]], [[John Archibald Wheeler]] (1970) ''Gravitation''. W.H. Freeman. {{isbn|0-7167-0344-0}}. A basic review of [[differential geometry]] in the special case of four-dimensional [[spacetime]].
* Steven Rosenberg (1997) ''The Laplacian on a Riemannian manifold''. Cambridge University Press. {{isbn|0-521-46831-0}}. An introduction to the [[heat equation]] and the [[Atiyah-Singer theorem]].
* [http://people.oregonstate.edu/~drayt/Courses/MTH434/2007/dual.pdf Tevian Dray (1999) ''The Hodge Dual Operator'']. A thorough overview of the definition and properties of the Hodge star operator.

{{Tensors}}

{{DEFAULTSORT:Hodge Dual}}
[[Category:Differential forms]]
[[Category:Riemannian geometry]]
[[Category:Duality theories]]