# 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**
, 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:

- .

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 is the same tensor as , 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

## References

- 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 0-679-77631-1.

## External links

- Index Gymnastics, Wolfram Alpha