Mixed tensor

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 $\binom$, 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

Consider the following octet of related tensors:
$T_, \ T_ ^\gamma, \ T_\alpha ^\beta _\gamma, \ T_\alpha ^, \ T^\alpha _, \ T^\alpha _\beta ^\gamma, \ T^ _\gamma, \ T^ .$
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 , and a given covariant index can be raised using the inverse metric tensor . Thus, could be called the ''index lowering operator'' and 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),
$T_ ^\lambda = T_ \, g^ ,$
where $T_ ^\lambda$ is the same tensor as $T_ ^\gamma$, because
$T_ ^\lambda \, \delta_\lambda ^\gamma = T_ ^\gamma,$
with Kronecker acting here like an identity matrix.

Likewise,
$T_\alpha ^\lambda _\gamma = T_ \, g^,$
$T_\alpha ^ = T_ \, g^ \, g^,$
$T^ _\gamma = g_ \, T^,$
$T^\alpha _ = g_ \, g_ \, T^.$

Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the Kronecker delta,
$g^ \, g_ = g^\mu _\nu = \delta^\mu _\nu ,$
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

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External links

Index Gymnastics
Wolfram Alpha

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Tensors