# 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 $\begin{pmatrix} M \\ N \end{pmatrix}$, 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+N-tuple of M one-forms and N vectors to a scalar.

## Index raising and lowering

Consider the following octet of related tensors:

$T_{\alpha \beta \gamma}, \ T_{\alpha \beta} {}^\gamma, \ T_\alpha {}^\beta {}_\gamma, \ T_\alpha {}^{\beta \gamma}, \ T^\alpha {}_{\beta \gamma}, \ T^\alpha {}_\beta {}^\gamma, \ T^{\alpha \beta} {}_\gamma, \ T^{\alpha \beta \gamma}$.

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 gμν, and a given covariant index can be raised using the inverse metric tensor gμν. Thus, gμν could be called the index lowering operator and gμν 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),

$T_{\alpha \beta} {}^\tau = T_{\alpha \beta \gamma} \, g^{\gamma \tau}$,

where Tαβτ is the same tensor as Tαβγ, because

$T_{\alpha \beta} {}^\tau \, \delta_\tau {}^\gamma = T_{\alpha \beta} {}^\gamma$,

with Kronecker δ acting here like an identity matrix.

Likewise,

$T_\alpha {}^\tau {}_\gamma = T_{\alpha \beta \gamma} \, g^{\beta \tau},$
$T_\alpha {}^{\tau \epsilon} = T_{\alpha \beta \gamma} \, g^{\beta \tau} \, g^{\gamma \epsilon},$
$T^{\alpha \beta} {}_\gamma = g_{\gamma \tau} \, T^{\alpha \beta \tau},$
$T^\alpha {}_{\tau \epsilon} = g_{\tau \beta} \, g_{\epsilon \gamma} \, T^{\alpha \beta \gamma}.$

Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the Kronecker delta,

$g^{\mu \lambda} \, g_{\lambda \nu} = g^\mu {}_\nu = \delta^\mu {}_\nu$,

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

## See also

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• mixed tensor — mišrusis tenzorius statusas T sritis fizika atitikmenys: angl. mixed tensor vok. gemischter Tensor, m rus. смешанный тензор, m pranc. tenseur mixte, m …   Fizikos terminų žodynas

• Tensor contraction — In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components… …   Wikipedia

• Tensor product — In mathematics, the tensor product, denoted by otimes, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. In each case the significance of the symbol is the same:… …   Wikipedia

• Glossary of tensor theory — This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:* Tensor * Classical treatment of tensors * Tensor (intrinsic definition) * Intermediate treatment of tensors * Application of tensor theory… …   Wikipedia

• Two-point tensor — Double vector redirects here. For dual vectors, see dual space. For bivectors, see bivector. Two point tensors, or double vectors, are tensor like quantities which transform as vectors with respect to each of their indices and are used in… …   Wikipedia

• gemischter Tensor — mišrusis tenzorius statusas T sritis fizika atitikmenys: angl. mixed tensor vok. gemischter Tensor, m rus. смешанный тензор, m pranc. tenseur mixte, m …   Fizikos terminų žodynas

• Stress-energy tensor — The stress energy tensor (sometimes stress energy momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of …   Wikipedia

• Covariance and contravariance of vectors — For other uses of covariant or contravariant , see covariance and contravariance. In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities… …   Wikipedia

• List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia

• Pullback (differential geometry) — Suppose that φ : M → N is a smooth map between smooth manifolds M and N ; then there is an associated linear map from the space of 1 forms on N (the linear space of sections of the cotangent bundle) to the space of 1 forms on M . This linear map… …   Wikipedia

### Share the article and excerpts

##### Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.