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 , 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 tensor can be defined as a linear function which maps an M+Ntuple of M oneforms and N vectors to a scalar.
Index raising and lowering
Consider the following octet of related tensors:
 .
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),
 ,
where T_{αβ}^{τ} is the same tensor as T_{αβ}^{γ}, because
 ,
with Kronecker δ acting here like an identity matrix.
Likewise,
Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the Kronecker delta,
 ,
so any mixed version of the metric tensor will be equal to the Kronecker delta, which will also be mixed.
See also
Categories: Tensors
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