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

A tensor **A** that is antisymmetric on indices and has the property that the contraction with a tensor **B** that is symmetric on indices and is identically 0.

For a general tensor **U** with components and a pair of indices and **U** has symmetric and antisymmetric parts defined as:

(symmetric part) (antisymmetric part).

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

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**,

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

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

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

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

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

- Antisymmetric matrix
- Exterior algebra – Algebraic construction used in multilinear algebra and geometry
- Levi-Civita symbol – Antisymmetric permutation object acting on tensors
- Ricci calculus – Tensor index notation for tensor-based calculations
- Symmetric tensor – Tensor invariant under permutations of vectors it acts on
- Symmetrization

- Penrose, Roger (2007).
*The Road to Reality*. Vintage books. ISBN 0-679-77631-1. - J.A. Wheeler; C. Misner; K.S. Thorne (1973).
*Gravitation*. W.H. Freeman & Co. pp. 85–86, §3.5. ISBN 0-7167-0344-0.

- Antisymmetric Tensor – mathworld.wolfram.com