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| In [[mathematics]], the '''Hodge star operator''' or '''Hodge dual''' is a significant [[linear map]] introduced in general by [[W. V. D. Hodge]]. It is defined on the [[exterior algebra]] of a finite-dimensional [[orientation (mathematics)|oriented]] [[inner product space]].
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| ==Dimensionalities and algebra==
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| Suppose that ''n'' is the dimensionality of the oriented inner product space and ''k'' is an integer such that {{nowrap|0 ≤ ''k'' ≤ ''n''}}, then the Hodge star operator establishes a one-to-one mapping from the space of [[p-vector|''k''-vectors]] and the space of (''n''−''k'')-vectors. The image of a ''k''-vector under this mapping is called the ''Hodge dual'' of the ''k''-vector. The former space, of ''k''-vectors, has dimensionality
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| :<math> {n \choose k} </math> | |
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| while the latter has dimensionality
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| :<math> {n \choose n - k}, </math>
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| and by the symmetry of the [[binomial coefficient]]s, these two dimensionalities are in fact equal. Two [[vector space]]s 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 ''k''-vectors. The 'natural' definition means that this duality relationship can play a geometrical role in theories.
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| The first interesting case is on three-dimensional [[Euclidean space]] ''V''. In this context the relevant row of [[Pascal's triangle]] reads
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| :1, 3, 3, 1
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| and the Hodge dual sets up an isomorphism between the two three-dimensional spaces, which are ''V'' itself and the space of [[wedge product]]s 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.
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| ==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 ''k''-vectors 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 [[Laplace-de Rham operator]], which leads to the [[Hodge decomposition]] of [[differential forms]] in the case of [[Compact space|compact]] [[Riemannian manifold]]s.
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| ==Formal definition of the Hodge star of ''k''-vectors==
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| The '''Hodge star operator''' on a [[vector space]] ''V'' with a [[nondegenerate]] [[symmetric bilinear form]] (herein aka ''inner product'') is a linear operator on the [[exterior algebra]] of ''V'', mapping ''k''-vectors to {{nowrap|(''n'' − ''k'')}}-vectors where {{nowrap|1=''n'' = dim ''V''}}, for {{nowrap|1=0 ≤ ''k'' ≤ ''n''}}. It has the following property, which defines it completely: given two ''k''-vectors α, β
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| :<math>\alpha \wedge (\star \beta) = \langle \alpha,\beta \rangle \omega</math>
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| where <math>\langle \cdot,\cdot \rangle</math> denotes the [[Exterior algebra#Inner product|inner product on ''k''-vectors]] and ω is the preferred unit ''n''-vector.
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| The inner product <math>\langle \cdot,\cdot \rangle</math> on ''k''-vectors is extended from that on ''V'' by requiring that <math>\langle \alpha,\beta \rangle = \det(\langle \alpha_i,\beta_j \rangle)</math> for any decomposable ''k''-vectors <math>\alpha = \alpha_1 \wedge \dots \wedge \alpha_k</math> and <math>\beta = \beta_1 \wedge \dots \wedge \beta_k</math>.
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| The unit ''n''-vector ω is unique up to a sign. The preferred choice of ω defines an [[orientation (mathematics)|orientation]] on ''V''.
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| ==Explanation==
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| Let ''W'' be a vector space, with an inner product <math>\langle\cdot, \cdot\rangle_W</math>. For every [[Continuous dual space#Dual space of a topological vector space|continuous]] linear functional <math>f \in W^*</math> there exists a unique vector ''v'' in ''W'' such that <math>f(w) = \langle w, v \rangle_W</math> for all ''w'' in ''W''. The map <math>W^* \to W</math> given by <math>f \mapsto v</math> is an isomorphism. This holds for ''all'' vector spaces with an inner product, and can be used to explain the Hodge dual.
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| Let ''V'' be an ''n''-dimensional vector space with basis <math>\{e_1,\ldots,e_n\}</math>. For 0 ≤ ''k''≤ ''n'', consider the exterior power spaces <math>\bigwedge^k V</math> and <math>\bigwedge^{n-k} V</math>. For each <math>\lambda \in \bigwedge^k V</math> and <math>\theta \in \bigwedge^{n-k} V</math>, we have <math>\lambda \wedge \theta \in \bigwedge^n V</math>. There is, up to a scalar, only one ''n''-vector, 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 \bigwedge^k V</math> and <math>\theta \in \bigwedge^{n-k} V</math>.
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| Consider a ''fixed'' <math>\lambda \in \bigwedge^k V</math>. There exists a unique linear function <math>f_{\lambda} \in \left(\bigwedge^{n-k} V\right)^{\! *}</math> such that <math>\lambda \wedge \theta = f_{\lambda}(\theta) \, (e_1\wedge\ldots\wedge e_n)</math> for all <math>\theta \in \bigwedge^{n-k} V</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 (''n''–''k'')-vectors]], then there exists a unique (''n''–''k'')-vector, say <math>\star \lambda \in \bigwedge^{n-k} V</math>, such that <math>f_{\lambda}(\theta) = \langle \theta, \star \lambda\rangle</math> for all <math>\theta \in \bigwedge^{n-k} V</math>. This (''n''–''k'')-vector <math>\star\lambda</math> is the Hodge dual of λ, and is the image of the <math>f_{\lambda}</math> under the canonical isomorphism between <math>\left(\bigwedge^{n-k} V\right)^{\! *}</math> and <math>\bigwedge^{n-k} V</math>. Thus, <math> \star : \bigwedge^{k} V \to \bigwedge^{n-k} V</math>.
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| ==Computation of the Hodge star==
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| Given an orthonormal basis <math>(e_1,e_2,\dots,e_n)</math> ordered such that <math>\omega = e_1\wedge e_2\wedge \cdots \wedge e_n</math>, we see that
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| :<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>
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| where <math>\{i_1, \cdots i_k, i_{k+1} \cdots i_n\}</math> is an even permutation of <math>\{1, 2, \cdots n\}</math>.
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| 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
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| :<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>
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| ==Index notation for the star operator==
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| Using index notation, the Hodge dual is obtained by contracting the indices of a ''k''-form with the ''n''-dimensional completely antisymmetric '''Levi-Civita tensor'''. This differs from the [[Levi-Civita symbol]] by a factor of |det ''g''|<sup>½</sup>, where ''g'' is an inner product (the [[metric tensor]]). The absolute value of the determinant is necessary if ''g'' is not positive-definite, e.g. for tangent spaces to [[Pseudo-Riemannian manifold#Lorentzian manifold|Lorentzian manifolds]].
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| Thus one writes<ref>The Geometry of Physics (3rd edition), T. Frankel, Cambridge University Press, 2012, ISBN 978-1107-602601</ref>
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| :<math>(\star \eta)_{i_1,i_2,\ldots,i_{n-k}} = \frac{1}{(k)!} \eta^{j_1,\ldots,j_k}\,\sqrt {|\det g|} \,\epsilon_{j_1,\ldots,j_k,i_1,\ldots,i_{n-k}}</math>
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| where ''η'' is an arbitrary antisymmetric [[tensor]] in ''k'' indices. It is understood that [[raising and lowering indices|indices are raised and lowered]] using the same inner product ''g'' as in the definition of the Levi-Civita 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 anti-symmetric Levi-Civita symbol.
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| ==Examples==
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| A common example of the star operator is the case {{nowrap|1=''n'' = 3}}, when it can be taken as the correspondence between the vectors and the [[skew-symmetric matrix|skew-symmetric 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 space|Euclidean]] '''R'''<sup>3</sup>, one easily finds that
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| :<math>\star \mathrm{d}x=\mathrm{d}y\wedge \mathrm{d}z</math>
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| :<math>\star \mathrm{d}y=\mathrm{d}z\wedge \mathrm{d}x</math>
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| :<math>\star \mathrm{d}z=\mathrm{d}x\wedge \mathrm{d}y</math>
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| where d''x'', d''y'' and d''z'' are the standard orthonormal differential [[one-form]]s on '''R'''<sup>3</sup>. The Hodge dual 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.
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| ===Three-dimensional example===
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| Applied to three dimensions, the Hodge dual provides an [[isomorphism]] between [[axial vector]]s and [[bivector]]s, so each axial vector '''a''' is associated with a bivector '''A''' and vice-versa, that is:<ref name=Lounesto>
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| {{cite book |title=Clifford Algebras and Spinors, ''Volume 286 of London Mathematical Society Lecture Note Series'' |author=Pertti Lounesto |url=http://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>
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| : <math>\mathbf{A} = \star \mathbf{a}\,,\quad\mathbf{a} = \star \mathbf{A} \ ,</math>
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| where <math>\star</math> indicates the dual operation. 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|''C''ℓ<sub>3</sub>(R)]],<ref name=Datta>{{cite book |title=Geometric algebra and applications to physics |chapter=The pseudoscalar and imaginary unit |url=http://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> ''i'' = '''e'''<sub>1</sub>'''e'''<sub>2</sub>'''e'''<sub>3</sub> (the vectors {{nowrap|{ '''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=http://books.google.com/books?id=oaoLbMS3ErwC&pg=PA100}}</ref>
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| : <math>\mathbf{A} = \mathbf{a}i\,,\quad\mathbf{a} = - \mathbf{A} i. </math>
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| The dual of a vector is obtained by multiplication by ''i'', as established using the properties of the [[Geometric_product#The_geometric_product|geometric product]] of the algebra as follows:
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| :<math>\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} \ </math>
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| ::<math>= 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 \ </math>
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| ::<math>= a_1 \mathbf{e_2 e_3} +a_2 \mathbf{e_3 e_1} +a_3 \mathbf{e_1 e_2} = (\star \mathbf a )\ ;</math>
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| and also, in the dual space spanned by {{nowrap|{ '''e'''<sub>ℓ</sub>'''e'''<sub>''m''</sub> }|}}:
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| :<math>\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} \ </math>
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| ::<math>= 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 \ </math>
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| ::<math>=-\left( A_1 \mathbf{e_1} + A_2 \mathbf{e_2} + A_3 \mathbf{e_3} \right) = - (\star \mathbf A )\ . </math>
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| In establishing these results, the identities are used:
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| :<math>(\mathbf{e_1e_2})^2 =\mathbf{e_1e_2e_1e_2}= -\mathbf{e_1e_2e_2e_1} = -1 \ , </math>
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| and:
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| :<math>\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 \ . </math>
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| These relations between the dual <math>\star</math> and ''i'' apply to any vectors. Here they are applied to relate the axial vector created as the [[cross product]] {{nowrap|1='''a''' = '''u''' × '''v'''}} to the bivector-valued [[exterior product]] {{nowrap|1='''A''' = '''u''' ∧ '''v'''}} of two [[polar vectors|polar]] (that is, not axial) vectors '''u''' and '''v'''; the two products can be written as [[determinant]]s expressed in the same way:
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| : <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}_{23} & \mathbf{e}_{31} & \mathbf{e}_{12}\\u_1 & u_2 & u_3\\v_1 & v_2 & v_3 \end{vmatrix}\ ,</math>
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| using the notation {{nowrap|1='''e'''<sub>ℓ''m''</sub> = '''e'''<sub>ℓ</sub>'''e'''<sub>''m''</sub>}}. These expressions show these two types of vector are Hodge duals:<ref name=Lounesto/>
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| :<math>\star (\mathbf u \wedge \mathbf v )=\mathbf {u \times v}\,,\quad\star (\mathbf u \times \mathbf v ) = \mathbf u \wedge \mathbf v \,,</math>
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| as a result of the relations:
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| :<math>\star \mathbf e_{\ell} = \mathbf e_{\ell} \mathit i =\mathbf e_{\ell} \mathbf{e_1e_2e_3} = \mathbf e_m \mathbf e_n \,, </math>
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| with <math>\ell, m, n</math> cyclic, | |
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| and:
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| :<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_1e_2e_3} =\mathbf e_{n} </math>
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| also with <math>\ell, m, n</math> cyclic.
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| Using the implementation of <math>\star</math> based upon ''i'', the commonly used relations are:<ref name=Hestenes>
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| {{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
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| |author=David Hestenes |url=http://books.google.com/books?id=AlvTCEzSI5wC&pg=PA60 |page=60 }}
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| </ref>
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| :<math> \mathbf {u \times v} = -(\mathbf u \wedge \mathbf v ) i \,,\quad \mathbf u \wedge \mathbf v = (\mathbf {u \times v} ) i \ . </math>
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| ===Four dimensions===
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| In case {{nowrap|1=''n'' = 4}}, the Hodge dual acts as an [[endomorphism]] of the second exterior power, of dimension 6. It is an [[involution (mathematics)|involution]], so it splits it into ''self-dual'' and ''anti-self-dual'' subspaces, on which it acts respectively as +1 and −1.
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| Another useful example is {{nowrap|1=''n'' = 4}} Minkowski spacetime with metric signature <math>(+, -, -, -)</math> and coordinates <math>(t, x, y, z)</math> where (using <math>\varepsilon_{0123} = 1</math>)
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| :<math>\star \mathrm{d}t=\mathrm{d}x\wedge \mathrm{d}y \wedge\mathrm{d}z</math>
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| :<math>\star \mathrm{d}x=\mathrm{d}t\wedge \mathrm{d}y \wedge\mathrm{d}z</math>
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| :<math>\star \mathrm{d}y=\mathrm{d}t\wedge \mathrm{d}z \wedge\mathrm{d}x</math>
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| :<math>\star \mathrm{d}z=\mathrm{d}t\wedge \mathrm{d}x \wedge\mathrm{d}y</math>
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| for [[one-form]]s while
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| :<math>\star \mathrm{d}t \wedge\mathrm{d}x = - \mathrm{d}y\wedge \mathrm{d}z</math>
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| :<math>\star \mathrm{d}t \wedge\mathrm{d}y = \mathrm{d}x\wedge \mathrm{d}z</math>
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| :<math>\star \mathrm{d}t \wedge\mathrm{d}z = - \mathrm{d}x\wedge \mathrm{d}y</math>
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| :<math>\star \mathrm{d}x \wedge\mathrm{d}y = \mathrm{d}t\wedge \mathrm{d}z</math>
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| :<math>\star \mathrm{d}x \wedge\mathrm{d}z = - \mathrm{d}t\wedge \mathrm{d}y</math>
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| :<math>\star \mathrm{d}y \wedge\mathrm{d}z = \mathrm{d}t\wedge \mathrm{d}x</math>
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| for [[two-form]]s.
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| ==Inner product of ''k''-vectors==
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| The Hodge dual induces an inner product on the space of ''k''-vectors, that is, on the [[exterior algebra]] of ''V''. Given two ''k''-vectors <math>\eta</math> and <math>\zeta</math>, one has
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| :<math>\zeta\wedge \star \eta = \langle\zeta, \eta \rangle\;\omega</math>
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| where ''ω'' is the normalised ''n''-form (i.e. {{nowrap|1=''ω'' ∧ ∗''ω'' = ''ω''}}). In the calculus of exterior [[differential form]]s on a [[pseudo-Riemannian manifold]] of dimension ''n'', the normalised ''n''-form is called the [[volume form]] and can be written as
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| :<math>\omega=\sqrt{|\det [g_{ij}]|}\;\mathrm{d}x^1\wedge\cdots\wedge \mathrm{d}x^n</math>
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| where <math>[g_{ij}]</math> is the matrix of components of the [[metric tensor]] on the manifold in the [[coordinate basis]].
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| If an inner product is given on <math> \Lambda^k(V) </math>, then this equation can be regarded as an alternative definition of the Hodge dual.<ref>{{cite book | last = Darling| first = R. W. R. | title = Differential forms and connections| publisher = Cambridge University Press| year = 1994 }}</ref> The wedge products of elements of an orthonormal basis in ''V'' form an orthonormal basis of the exterior algebra of ''V''.
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| ==Duality==
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| 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 ''k''-vector <math>\eta \in \Lambda^k (V)</math> in an ''n''-dimensional space ''V'', one has
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| :<math>\star {\star \eta}=(-1)^{k(n-k)}s\eta</math>
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| where ''s'' is related to the [[metric signature|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
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| (+,−,−,−) or (−,+,+,+) then ''s''=−1. For ordinary Euclidean spaces, the signature is always positive, and so ''s''=+1. When the Hodge star is extended to pseudo-Riemannian manifolds, then the above inner product is understood to be the metric in diagonal form.
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| Note that the above identity implies that the inverse of <math>\star</math> can be given as
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| :<math> \star^{-1}:\Lambda^k\ni\eta \mapsto (-1)^{k(n-k)}s{\star \eta} \in\Lambda^{n-k}</math>
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| 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''.
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| Therefore, if ''n'' is odd, it holds for any ''k'' that
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| :<math> \star^{-1} = s\star </math>
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| whereas, if ''n'' is even, it holds that
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| :<math> \star^{-1} = (-1)^k s\star</math>
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| where k is the degree of the forms operated on.
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| ==Hodge star on manifolds==
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| One can repeat the construction above for each [[cotangent space]] of an ''n''-dimensional oriented [[Riemannian manifold|Riemannian]] or [[pseudo-Riemannian manifold]], and get the '''Hodge dual''' (''n''−''k'')-form, of a [[differential form|''k''-form]]. The Hodge star then induces an [[Lp space|L<sup>2</sup>-norm]] inner product on the differential forms on the manifold. One writes
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| :<math>(\eta,\zeta)=\int_M \eta\wedge \star \zeta = \int_M \langle \eta, \zeta \rangle \; \mathrm{d} \text{Vol} </math>
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| for the inner product of [[fibre bundle|sections]] <math>\eta</math> and <math>\zeta</math> of <math>\Lambda^k(T^*M)</math>. (The set of sections is frequently denoted as <math>\Omega^k(M)=\Gamma(\Lambda^k(T^*M))</math>. Elements of <math>\Omega^k(M)</math> are called exterior k-forms).
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| More generally, in the non-oriented case, one can define the hodge star of a ''k''-form as a (''n''−''k'')-[[pseudotensor|pseudo differential form]]; that is, a differential forms with values in the [[Canonical bundle|canonical line bundle]].
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| ===The codifferential===<!-- This section is linked from [[Differential form]] -->
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| The most important application of the Hodge dual on manifolds to is to define the '''codifferential''' δ. Let
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| :<math>\delta = (-1)^{nk + n + 1}s\, {\star \mathrm{d}\star} = (-1)^k\,{\star^{-1}\mathrm{d}\star} </math> | |
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| where ''d'' is the [[exterior derivative]] or differential, and ''s''=+1 for Riemannian manifolds.
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| :<math>\mathrm{d}:\Omega^k(M)\rightarrow \Omega^{k+1}(M)</math>
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| while
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| :<math>\delta:\Omega^k(M)\rightarrow \Omega^{k-1}(M).</math>
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| The codifferential is not an [[antiderivation]] on the exterior algebra,
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| in contrast to the exterior derivative.
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| The codifferential is the adjoint of the exterior derivative, in that
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| :<math> \langle \eta,\delta \zeta\rangle = \langle \mathrm{d}\eta,\zeta\rangle. </math>
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| where ''ζ'' is a (''k''+1)-form and ''η'' a ''k''-form. This identity follows from Stokes' theorem for smooth forms, when
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| :<math>\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}})=\langle \mathrm{d}\eta,\zeta\rangle -\langle\eta,\delta\zeta\rangle</math>
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| i.e. when <math>M</math> has empty boundary or when <math>\eta</math> 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).
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| Notice that since the differential satisfies <math> \mathrm{d}^2=0 </math>, the codifferential has the corresponding property
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| :<math>\! \delta^2 = s^2{\star \mathrm{d}{\star {\star \mathrm{d}{\star}}}} = (-1)^{k(n-k)} s^3{\star \mathrm{d}^2\star} = 0 </math>
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| The [[Laplace-Beltrami operator|Laplace-deRham]] operator is given by
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| :<math>\! \Delta=(\delta+\mathrm{d})^2 = \delta \mathrm{d} + \mathrm{d}\delta</math>
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| and lies at the heart of [[Hodge theory]]. It is symmetric:
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| :<math>\langle\Delta \zeta,\eta\rangle = \langle\zeta,\Delta \eta\rangle</math>
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| and non-negative:
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| :<math>\langle\Delta\eta,\eta\rangle \ge 0.</math>
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| 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 ''k''-forms, and so the Hodge star induces an isomorphism of cohomology groups
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| :<math>\star : H^k_\Delta(M)\to H^{n-k}_\Delta(M),</math>
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| which in turn gives canonical identifications via [[Poincaré duality]] of ''H''<sup>k</sup>(''M'') with its [[dual space]].
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| ==Derivatives in three dimensions==
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| The combination of the <math>\star</math> operator and the [[exterior derivative]] ''d'' generates the classical operators [[gradient|grad]], [[Curl (mathematics)|curl]], and [[divergence|div]], in three dimensions. This works out as follows: ''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:
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| :<math>\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.</math>
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| The second case followed by <math>\star</math> is an operator on 1-forms (<math>\eta=A\,\mathrm{d}x+B\,\mathrm{d}y+C\,\mathrm{d}z</math>) that in components is the curl operator:
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| :<math>\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.</math>
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| Applying the Hodge star gives:
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| :<math>\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.</math>
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| The final case prefaced and followed by <math>\star</math>, takes a 1-form (<math>\eta=A\,\mathrm{d}x+B\,\mathrm{d}y+C\,\mathrm{d}z</math>) to a 0-form (function); written out in components it is the divergence operator:
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| :<math>\star\eta=A\,\mathrm{d}y\wedge \mathrm{d}z-B\,\mathrm{d}x\wedge \mathrm{d}z+C\,\mathrm{d}x\wedge \mathrm{d}y</math>
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| :<math>\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</math>
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| :<math>\star \mathrm{d}{\star\eta}=\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}.</math>
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| One advantage of this expression is that the identity <math>\mathrm{d}^2=0</math>, which is true in all cases, sums up two others, namely that <math>\operatorname{curl}(\operatorname{grad}(f))=0</math> and <math>\operatorname{div}(\operatorname{curl}(\mathbf{F}))=0</math>. 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.
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| One can also obtain the [[Laplacian]]. Using the information above and the fact that <math>\Delta f = \operatorname{div}\; \operatorname{grad} f</math> then for a 0-form, <math>\omega=f(x,y,z)</math>:
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| :<math> \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}</math>
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * 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.
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| * 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.
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| * [[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]].
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| * 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]].
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| * [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 dual operator.
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| {{Tensors}}
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| {{DEFAULTSORT:Hodge Dual}}
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| [[Category:Differential forms]]
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| [[Category:Riemannian geometry]]
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| [[Category:Duality theories]]
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