Liu Hui

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 $\scriptstyle {\binom {M}{N}}$, 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

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Consider the following octet of related tensors:

$T_{\alpha \beta \gamma },\ T_{\alpha \beta }{}^{\gamma },\ T_{\alpha }{}^{\beta }{}_{\gamma },\ T_{\alpha }{}^{\beta \gamma },\ T^{\alpha }{}_{\beta \gamma },\ T^{\alpha }{}_{\beta }{}^{\gamma },\ T^{\alpha \beta }{}_{\gamma },\ T^{\alpha \beta \gamma }$.

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 the metric tensor g_μν, and a given covariant index can be raised using the inverse metric tensor g^μν. Thus, g_μν could be called the index lowering operator and 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),

$T_{\alpha \beta }{}^{\lambda }=T_{\alpha \beta \gamma }\,g^{\gamma \lambda }$,

where $T_{\alpha \beta }{}^{\lambda }$ is the same tensor as $T_{\alpha \beta }{}^{\gamma }$, because

$T_{\alpha \beta }{}^{\lambda }\,\delta _{\lambda }{}^{\gamma }=T_{\alpha \beta }{}^{\gamma }$,

with Kronecker δ acting here like an identity matrix.

Likewise,

$T_{\alpha }{}^{\lambda }{}_{\gamma }=T_{\alpha \beta \gamma }\,g^{\beta \lambda },$

$T_{\alpha }{}^{\lambda \epsilon }=T_{\alpha \beta \gamma }\,g^{\beta \lambda }\,g^{\gamma \epsilon },$

$T^{\alpha \beta }{}_{\gamma }=g_{\gamma \lambda }\,T^{\alpha \beta \lambda },$

$T^{\alpha }{}_{\lambda \epsilon }=g_{\lambda \beta }\,g_{\epsilon \gamma }\,T^{\alpha \beta \gamma }.$

Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the Kronecker delta,

$g^{\mu \lambda }\,g_{\lambda \nu }=g^{\mu }{}_{\nu }=\delta ^{\mu }{}_{\nu }$,

so any mixed version of the metric tensor will be equal to the Kronecker delta, which will also be mixed.

See also

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

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• Index Gymnastics, Wolfram Alpha