mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

, a

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

is antisymmetric or alternating on (or with respect to) an index subset if it alternates

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...

(+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally either be all ''covariant'' or all ''contravariant''. For example,

T_ = -T_ = T_ = -T_ = T_ = -T_

holds when the tensor is antisymmetric with respect to its first three indices. If a tensor changes sign under exchange of ''each'' pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor field of order

k

may be referred to as a differential

k

-form, and a completely antisymmetric contravariant tensor field may be referred to as a

k

-vector field.

Antisymmetric and symmetric tensors

A tensor A that is antisymmetric on indices

i

and

j

has the property that the contraction with a tensor B that is symmetric on indices

i

and

j

is identically 0. For a general tensor U with components

U_

and a pair of indices

i

and

j,

U has symmetric and antisymmetric parts defined as: : Similar definitions can be given for other pairs of indices. As the term "part" suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in

U_ = U_ + U_.

Notation

A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions, for an order 2 covariant tensor M,

M_ = \frac(M_ - M_),

and for an order 3 covariant tensor T,

T_ = \frac(T_-T_+T_-T_+T_-T_).

In any 2 and 3 dimensions, these can be written as

T_ &= \frac \, \delta_^ T_ . \end

where

\delta_^

is the generalized Kronecker delta, and the Einstein summation convention is in use. More generally, irrespective of the number of dimensions, antisymmetrization over

p

indices may be expressed as

T_ = \frac \delta_^ T_.

In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as:

T_ = \frac(T_ + T_) + \frac(T_ - T_).

This decomposition is not in general true for tensors of rank 3 or more, which have more complex symmetries.

Examples

Totally antisymmetric tensors include: * Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric). * The electromagnetic tensor,

F_

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. * The Riemannian volume form on a

pseudo-Riemannian manifold In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

Antisymmetric and symmetric tensors

Notation

Examples

See also

Notes

References

External links