# Antisymmetric tensor

In mathematics and theoretical physics, a tensor is **antisymmetric on** (or **with respect to**) **an index subset** if it alternates sign (+/−) when any two indices of the subset are interchanged.^{[1]}^{[2]} The index subset must generally either be all *covariant* or all *contravariant*.

For example,

holds when the tensor is antisymmetric on it first three indices.

If a tensor changes sign under exchange of *any* pair of its indices, then the tensor is **completely** (or **totally**) **antisymmetric**. A completely antisymmetric covariant tensor of order *p* may be referred to as a *p*-form, and a completely antisymmetric contravariant tensor may be referred to as a *p*-vector.

## 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 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

## 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**,

and for an order 3 covariant tensor **T**,

In any number of dimensions, these are equivalent to

More generally, irrespective of the number of dimensions, antisymmetrization over *p* indices may be expressed as

In the above,

is the generalized Kronecker delta of the appropriate order.

## Examples

Antisymmetric tensors include:

- The electromagnetic tensor, in electromagnetism
- The Riemannian volume form on a pseudo-Riemannian manifold.

## See also

## References

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## External links

- [1] - mathworld, wolfram