tensor analysis In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...

, a mixed tensor is a

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other 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

\binom

, 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 In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...

which maps an (''M'' + ''N'')-tuple of ''M''

one-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to e ...

s and ''N''

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

s to a scalar.

Changing the tensor type

Consider the following octet of related tensors:

T_, \ T_ ^\gamma, \ T_\alpha ^\beta _\gamma, \ 
T_\alpha ^, \ T^\alpha _, \ T^\alpha _\beta ^\gamma, \ 
T^ _\gamma, \ T^ .

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 , and a given covariant index can be raised using the inverse metric tensor . Thus, could be called the ''index lowering operator'' and 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_ ^\lambda = T_ \, g^ ,

where

T_ ^\lambda

is the same tensor as

T_ ^\gamma

, because

T_ ^\lambda \, \delta_\lambda ^\gamma = T_ ^\gamma,

with Kronecker acting here like an identity matrix. Likewise,

T_\alpha ^\lambda _\gamma = T_ \, g^,

T_\alpha ^ = T_ \, g^ \, g^,

T^ _\gamma = g_ \, T^,

T^\alpha _ = g_ \, g_ \, T^.

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

Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...

g^ \, g_ = 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.

References

* * *

External links

Index Gymnastics
Wolfram Alpha {{DEFAULTSORT:Mixed Tensor Tensors

Changing the tensor type

Examples

See also

References

External links