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 covariant nor strictly 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 function which maps an (''M'' + ''N'')-tuple of ''M'' one-forms and ''N'' vectors to a 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''''μν''}}, and a given covariant index can be raised using the inverse metric tensor {{math|''g''''μν''}}. Thus, {{math|''g''''μν''}} could be called the ''index lowering operator'' and {{math|''g''''μν''}} 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.

==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 |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

{{DEFAULTSORT:Mixed Tensor}} Category:Tensors