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${\textstyle {\binom {M}{N}}}$, 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 Mone-forms and Nvectors to a scalar.

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 tensorg_{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).

ExamplesEdit

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),