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| 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).
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| A mixed tensor of '''type''' or '''valence''' <math>\scriptstyle\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]].
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| ==Changing the tensor type==
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| {{main|raising and lowering indices}}
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| Consider the following octet of related tensors:
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| :<math> T_{\alpha \beta \gamma}, \ T_{\alpha \beta} {}^\gamma, \ T_\alpha {}^\beta {}_\gamma, \
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| T_\alpha {}^{\beta \gamma}, \ T^\alpha {}_{\beta \gamma}, \ T^\alpha {}_\beta {}^\gamma, \
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| T^{\alpha \beta} {}_\gamma, \ T^{\alpha \beta \gamma} </math>.
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| 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]] ''g''<sub>μν</sub>, and a given covariant index can be raised using the inverse metric tensor ''g''<sup>μν</sup>. Thus, ''g''<sub>μν</sub> could be called the ''index lowering operator'' and ''g<sup>μν</sup>'' the ''index raising operator''.
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| 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).
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| ===Examples===
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| 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),
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| :<math> T_{\alpha \beta} {}^\lambda = T_{\alpha \beta \gamma} \, g^{\gamma \lambda} </math>,
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| where <math> T_{\alpha \beta} {}^\lambda </math> is the same tensor as <math> T_{\alpha \beta} {}^\gamma </math>, because
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| :<math> T_{\alpha \beta} {}^\lambda \, \delta_\lambda {}^\gamma = T_{\alpha \beta} {}^\gamma </math>,
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| with Kronecker ''δ'' acting here like an identity matrix.
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| Likewise,
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| :<math> T_\alpha {}^\lambda {}_\gamma = T_{\alpha \beta \gamma} \, g^{\beta \lambda}, </math> | |
| :<math> T_\alpha {}^{\lambda \epsilon} = T_{\alpha \beta \gamma} \, g^{\beta \lambda} \, g^{\gamma \epsilon},</math> | |
| :<math> T^{\alpha \beta} {}_\gamma = g_{\gamma \lambda} \, T^{\alpha \beta \lambda},</math>
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| :<math> T^\alpha {}_{\lambda \epsilon} = g_{\lambda \beta} \, g_{\epsilon \gamma} \, T^{\alpha \beta \gamma}. </math>
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| Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the [[Kronecker delta]],
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| :<math> g^{\mu \lambda} \, g_{\lambda \nu} = g^\mu {}_\nu = \delta^\mu {}_\nu </math>,
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| so any mixed version of the metric tensor will be equal to the Kronecker delta, which will also be mixed.
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| ==See also==
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| * [[Covariance and contravariance of vectors]]
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| * [[Einstein notation]]
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| * [[Ricci calculus]]
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| * [[Tensor (intrinsic definition)]]
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| * [[Two-point tensor]]
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| ==References==
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| * {{cite book |author=D.C. Kay| title=Tensor Calculus| publisher= Schaum’s Outlines, McGraw Hill (USA)| year=1988 | isbn=0-07-033484-6}}
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| * {{cite book |page=p.85-86, §3.5| author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}
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| * {{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books| year=2007 | isbn=0-679-77631-1}}
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| ==External links==
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| * [http://mathworld.wolfram.com/IndexGymnastics.html Index Gymnastics], Wolfram Alpha
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| {{tensors}}
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| {{DEFAULTSORT:Mixed Tensor}}
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| [[Category:Tensors]]
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My person who wrote each article is called Leland but it's not this most masucline name . Managing people is without question where his primary sales comes from. His wife and him live on Massachusetts and he enjoys everything that he conditions there. Base jumping is something that he's been doing for months. He 's running and maintaining a blog here: http://Prometeu.net
My web page - clash of clans hacks