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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:## See also

- $T\_\{ijkdots\}\; =\; -T\_\{jikdots\}$

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 $U\_\{ijkdots\}$ and a pair of indices i and j, U has symmetric and antisymmetric parts defined as:

- $U\_\{(ij)kdots\}=frac\{1\}\{2\}(U\_\{ijkdots\}+U\_\{jikdots\})$ (symmetric part)

- $U\_\{[ij]kdots\}=frac\{1\}\{2\}(U\_\{ijkdots\}-U\_\{jikdots\})$ (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 $U\_\{ijkdots\}=U\_\{(ij)kdots\}+U\_\{[ij]kdots\}$

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

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Last updated on Monday August 25, 2008 at 11:31:40 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday August 25, 2008 at 11:31:40 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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