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Intensor analysis, amixed tensor is atensor which is neither strictlycovariant nor strictlycontravariant; 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 oftype orvalence$(\genfrac{}{}{0ex}{}{M}{N})$, also written "type (M,N)", with bothM > 0 andN > 0, is a tensor which hasM contravariant indices andN covariant indices. Such a tensor can be defined as alinear function which maps an (M +N)-tuple ofMone-forms andNvectors to ascalar.

Consider the following octet of related tensors:$${\displaystyle T_{\alpha \beta \gamma },T_{\alpha \beta }{}^{\gamma },T_{\alpha }{}^{\beta }{}_{\gamma },T_{\alpha }{}^{\beta \gamma },T^{\alpha }{}_{\beta \gamma },T^{\alpha }{}_{\beta }{}^{\gamma },T^{\alpha \beta }{}_{\gamma },T^{\alpha \beta \gamma }.}$$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 themetric tensorg_μν, and a given covariant index can be raised using the inverse metric tensorg^μν. Thus,g_μν could be called theindex lowering operator andg^μν theindex 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).

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),$${\displaystyle T_{\alpha \beta }{}^{\lambda }=T_{\alpha \beta \gamma }{g}^{\gamma \lambda },}$$where${\displaystyle T_{\alpha \beta }{}^{\lambda }}$ is the same tensor as${\displaystyle T_{\alpha \beta }{}^{\gamma }}$, because$${\displaystyle T_{\alpha \beta }{}^{\lambda }{\delta }_{\lambda }{}^{\gamma }=T_{\alpha \beta }{}^{\gamma },}$$with Kroneckerδ acting here like an identity matrix.

Likewise,$${\displaystyle T_{\alpha }{}^{\lambda }{}_{\gamma }=T_{\alpha \beta \gamma }{g}^{\beta \lambda },}$$$${\displaystyle T_{\alpha }{}^{\lambda ϵ}=T_{\alpha \beta \gamma }{g}^{\beta \lambda }{g}^{\gamma ϵ},}$$$${\displaystyle T^{\alpha \beta }{}_{\gamma }=g_{\gamma \lambda }{T}^{\alpha \beta \lambda },}$$$${\displaystyle T^{\alpha }{}_{\lambda ϵ}=g_{\lambda \beta }{g}_{ϵ\gamma }{T}^{\alpha \beta \gamma }.}$$

Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding theKronecker delta,$${\displaystyle g^{\mu \lambda }{g}_{\lambda \nu }=g^{\mu }{}_{\nu }={\delta }^{\mu }{}_{\nu },}$$so any mixed version of the metric tensor will be equal to the Kronecker delta, which will also be mixed.

• Covariance and contravariance of vectors
• Einstein notation
• Ricci calculus
• Tensor (intrinsic definition)
• Two-point tensor

• D.C. Kay (1988).Tensor Calculus. Schaum’s Outlines, McGraw Hill (USA).ISBN 0-07-033484-6.
• Wheeler, J.A.; Misner, C.; Thorne, K.S. (1973). "§3.5 Working with Tensors".Gravitation. W.H. Freeman & Co. pp. 85–86.ISBN 0-7167-0344-0.
• R. Penrose (2007).The Road to Reality. Vintage books.ISBN 978-0-679-77631-4.

• Index Gymnastics, Wolfram Alpha

• Tensors

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