Editing Hodge star operator (section)

=== Three dimensions ===
A common example of the Hodge star operator is the case {{math|1=''n'' = 3}}, when it can be taken as the correspondence between vectors and  bivectors. Specifically, for [[Euclidean space|Euclidean]] '''R'''<sup>3</sup> with the basis <math>dx, dy, dz</math> of [[one-form]]s often used in [[vector calculus]], one finds that
<math display="block">\begin{align}
  {\star} \,dx &= dy \wedge dz \\
  {\star} \,dy &= dz \wedge dx \\
  {\star} \,dz &= dx \wedge dy.
\end{align}</math>

The Hodge star relates the exterior and cross product in three dimensions:<ref name="Lounesto" /> <math display="block">{\star} (\mathbf{u} \wedge \mathbf{v}) = \mathbf{u} \times \mathbf{v} \qquad {\star} (\mathbf{u} \times \mathbf {v}) = \mathbf{u} \wedge \mathbf{v} .</math> Applied to three dimensions, the Hodge star provides an [[isomorphism]] between [[axial vector]]s and [[bivector]]s, so each axial vector {{math|'''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 |chapter-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}, \ \ \mathbf{a} = {\star} \mathbf{A}</math>.

The Hodge star can also be interpreted as a form of the geometric correspondence between an [[axis of rotation]] and an [[infinitesimal rotation]] (see also: [[3D rotation group#Lie algebra]]) around the axis, with speed equal to the length of the axis of rotation. A scalar product on a vector space <math>V</math> gives an [[Dual space#Bilinear products and dual spaces|isomorphism]] <math>V\cong V^*\!</math>  identifying <math>V</math> with its [[dual space]], and the vector space <math>L(V,V)</math> is naturally isomorphic to the [[tensor product]] <math>V^*\!\!\otimes V\cong V\otimes V</math>. Thus for <math>V = \mathbb{R}^3</math>, the star mapping <math display="inline">\textstyle {\star} : V\to\bigwedge^{\!2}\! V \subset V\otimes V</math> takes each vector <math>\mathbf{v}</math> to a bivector <math>{\star} \mathbf{v} \in V\otimes V</math>, which corresponds to a linear operator <math>L_{\mathbf{v}} : V\to V</math>. Specifically, <math>L_{\mathbf{v}}</math> is a [[Skew-symmetric matrix|skew-symmetric]] operator, which corresponds to an infinitesimal rotation: that is, the macroscopic rotations around the axis <math>\mathbb{v}</math> are given by the [[matrix exponential]] <math>\exp(t L_{\mathbf{v}})</math>. With respect to the basis <math>dx, dy, dz</math> of <math>\R^3</math>, the tensor <math>dx\otimes dy</math> corresponds to a coordinate matrix with 1 in the <math>dx</math> row and <math>dy</math> column, etc., and the wedge <math>dx\wedge dy \,=\, dx\otimes dy - dy\otimes dx</math> is the skew-symmetric matrix <math>\scriptscriptstyle\left[\begin{array}{rrr}
\,0\!\! & \!\!1 & \!\!\!\!0\!\!\!\!\!\! 
\\[-.5em]
\,\!-1\!\!&\!\!0\!\!&\!\!\!\!0\!\!\!\!\!\! 
\\[-.5em]
\,0\!\! & \!\!0\!\! & \!\!\!\!0\!\!\!\!\!\!
\end{array}\!\!\!\right]</math>, etc. That is, we may interpret the star operator as: <math display="block"> \mathbf{v} = a\,dx + b\,dy + c\,dz
\quad\longrightarrow \quad 
{\star}{\mathbf{v}}
 \ \cong\ L_{\mathbf{v}} 
\ = \left[\begin{array}{rrr} 
0 & c & -b \\
-c & 0 & a \\
b & -a & 0
\end{array}\right].</math> 
Under this correspondence, cross product of vectors corresponds to the commutator [[Lie algebra|Lie bracket]] of linear operators: <math>L_{\mathbf{u}\times\mathbf{v}} = L_{\mathbf{v}} L_{\mathbf{u}} - L_{\mathbf{u}} L_{\mathbf{v}}=-\left[L_{\mathbf{u}}, L_{\mathbf{v}}\right]</math>.
<!-- 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|1=''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 display="block">\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 display="block">\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>}<nowiki/>}}:
<math display="block">\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 display="block">(\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 display="block">\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 display="block">\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 display="block">{\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 display="block">{\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 display="block">{\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 |author-link = David Hestenes|author=David Hestenes |url=https://books.google.com/books?id=AlvTCEzSI5wC&pg=PA60 |page=60 }}</ref>
<math display="block"> \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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