Editing Mixed tensor

{{Short description|Tensor having both covariant and contravariant indices}}
{{redirect|Tensor type|the array data type|Tensor type (computing)}}
{{No footnotes|date=October 2021}}
In [[tensor analysis]], a '''mixed tensor''' is a [[tensor]] which is neither strictly [[Covariance and contravariance of vectors|covariant]] nor strictly [[Covariance and contravariance of vectors|contravariant]]; at least one of the indices of a mixed tensor will be a subscript (covariant) and at least one of the indices will be a superscript (contravariant).

A mixed tensor of '''type''' or '''valence''' <math display="inline">\binom{M}{N}</math>, also written "type (''M'', ''N'')", with both ''M'' > 0 and ''N'' > 0, is a tensor which has ''M'' contravariant indices and ''N'' covariant indices.  Such a tensor can be defined as a [[linear operator|linear function]] which maps an (''M'' + ''N'')-tuple of ''M'' [[one-form]]s and ''N'' [[Vector (geometry)|vector]]s to a [[scalar (mathematics)|scalar]].

==Changing the tensor type==
{{main|Raising and lowering indices}}
Consider the following octet of related tensors:
<math display="block"> T_{\alpha \beta \gamma}, \ T_{\alpha \beta} {}^\gamma, \ T_\alpha {}^\beta {}_\gamma, \ 
T_\alpha {}^{\beta \gamma}, \ T^\alpha {}_{\beta \gamma}, \ T^\alpha {}_\beta {}^\gamma, \ 
T^{\alpha \beta} {}_\gamma, \ T^{\alpha \beta \gamma} .</math>
The first one is covariant, the last one contravariant, and the remaining ones mixed.  Notationally, these tensors differ from each other by the covariance/contravariance of their indices.  A given contravariant index of a tensor can be lowered using the [[metric tensor]] {{math|''g''<sub>''μν''</sub>}}, and a given covariant index can be raised using the inverse metric tensor {{math|''g''<sup>''μν''</sup>}}.  Thus, {{math|''g''<sub>''μν''</sub>}} could be called the ''index lowering operator'' and {{math|''g''<sup>''μν''</sup>}} the ''index raising operator''.

Generally, the covariant metric tensor, contracted with a tensor of type (''M'', ''N''), yields a tensor of type (''M'' − 1, ''N'' + 1), whereas its contravariant inverse, contracted with a tensor of type (''M'', ''N''), yields a tensor of type (''M'' + 1, ''N'' − 1).

===Examples===
As an example, a mixed tensor of type (1, 2) can be obtained by raising an index of a covariant tensor of type (0, 3),
<math display="block"> T_{\alpha \beta} {}^\lambda = T_{\alpha \beta \gamma} \, g^{\gamma \lambda} ,</math>
where <math> T_{\alpha \beta} {}^\lambda </math> is the same tensor as <math> T_{\alpha \beta} {}^\gamma </math>, because
<math display="block"> T_{\alpha \beta} {}^\lambda \, \delta_\lambda {}^\gamma = T_{\alpha \beta} {}^\gamma, </math>
with Kronecker {{math|''δ''}} acting here like an identity matrix.

Likewise,
<math display="block"> T_\alpha {}^\lambda {}_\gamma = T_{\alpha \beta \gamma} \, g^{\beta \lambda}, </math>
<math display="block"> T_\alpha {}^{\lambda \epsilon} = T_{\alpha \beta \gamma} \, g^{\beta \lambda} \, g^{\gamma \epsilon},</math>
<math display="block"> T^{\alpha \beta} {}_\gamma = g_{\gamma \lambda} \, T^{\alpha \beta \lambda},</math>
<math display="block"> T^\alpha {}_{\lambda \epsilon} = g_{\lambda \beta} \, g_{\epsilon \gamma} \, T^{\alpha \beta \gamma}. </math>

Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the [[Kronecker delta]],
<math display="block"> g^{\mu \lambda} \, g_{\lambda \nu} = g^\mu {}_\nu = \delta^\mu {}_\nu ,</math>
so any mixed version of the metric tensor will be equal to the Kronecker delta, which will also be mixed.

==See also==
* [[Covariance and contravariance of vectors]]
* [[Einstein notation]]
* [[Ricci calculus]]
* [[Tensor (intrinsic definition)]]
* [[Two-point tensor]]

==References==

* {{cite book |author=D.C. Kay| title=Tensor Calculus| publisher= Schaum’s Outlines, McGraw Hill (USA)| year=1988 | isbn=0-07-033484-6}}
* {{cite book |first1=J.A. |last1=Wheeler |first2=C. |last2=Misner |first3=K.S. |last3=Thorne |chapter=§3.5 Working with Tensors |title=[[Gravitation (book)|Gravitation]] |pages=85–86 |publisher=W.H. Freeman & Co |year=1973 |isbn=0-7167-0344-0}}
* {{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books| year=2007 | isbn=978-0-679-77631-4}}

==External links==

* [http://mathworld.wolfram.com/IndexGymnastics.html Index Gymnastics], Wolfram Alpha

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[[Category:Tensors]]