Definitions

# Antisymmetric tensor

In mathematics and theoretical physics, a tensor is antisymmetric on two indices i and j if it flips sign when the two indices are interchanged:

$T_\left\{ijkdots\right\} = -T_\left\{jikdots\right\}$

An antisymmetric tensor is a tensor for which there are two indices on which it is antisymmetric. If a tensor changes sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form.

A tensor A which is antisymmetric on indices i and j has the property that the contraction with a tensor B, which is symmetric on indices i and j, is identically 0.

For a general tensor U with components $U_\left\{ijkdots\right\}$ and a pair of indices i and j, U has symmetric and antisymmetric parts defined as:

$U_\left\{\left(ij\right)kdots\right\}=frac\left\{1\right\}\left\{2\right\}\left(U_\left\{ijkdots\right\}+U_\left\{jikdots\right\}\right)$ (symmetric part)

$U_\left\{\left[ij\right]kdots\right\}=frac\left\{1\right\}\left\{2\right\}\left(U_\left\{ijkdots\right\}-U_\left\{jikdots\right\}\right)$ (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 $U_\left\{ijkdots\right\}=U_\left\{\left(ij\right)kdots\right\}+U_\left\{\left[ij\right]kdots\right\}$

An important antisymmetric tensor in physics is the electromagnetic tensor F in electromagnetism.