In tensor analysis, a mixed tensor is a tensor which is neither covariant nor 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). For more technical Wiki articles on tensors, see the section later in this article. ... In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalar, vector and linear operator. ... In category theory, see covariant functor. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ...

A mixed tensor of type , 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+N-tuple of M one-forms and N vectors to a scalar. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... A one-form is a linear function which maps each vector in a vector space to a real number, such that the mapping is invariant with respect to coordinate transformations of the vector space. ... The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ... Scalar is a concept that has meaning in mathematics, physics, and computing. ...

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 difffer 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. In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...

Generally, the covariant metric tensor, contracted with a tensor of type , yields a tensor of type , whereas its contravariant inverse, contracted with a tensor of type , yields a tensor of type .

Examples

As an example, a mixed tensor of type can be obtained by raising an index of a covariant tensor of type ,

,

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, In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

,

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

See also

• Tensor (intrinsic definition)