Антисиметричний тензор

Тензор називається антисиметричним за двома індексами i та j, якщо він змінює знак при перестановці цих індексів:

T_{ijk\dots }=-T_{jik\dots }

Якщо тензор змінює знак при перестановці будь-якої пари індексів то такий тензор називається абсолютно антисиметричним тензором.

Для будь-якого тензора U, з компонентами $U_{ijk\dots }$ , можна побудувати симетричний і антисиметричний тензор за правилом:

$U_{(ij)k\dots }=(1/2)(U_{ijk\dots }+U_{jik\dots })$ (симетрична частина),

$U_{[ij]k\dots }=(1/2)(U_{ijk\dots }-U_{jik\dots })$ (антисиметрична частина),

аналогічно для інших індексів.

Під терміном «частина» йдеться про те, що $U_{ijk\dots }=U_{(ij)k\dots }+U_{[ij]k\dots }$

Властивості

Згортка тензора A, що є антисиметричним за індексами i і j з тензором B, що є симетричним за індексами i та j, рівна нулю. Доведення:

$A_{(ij)k\dots }B_{[ij]k\dots }=A_{(ji)k\dots }B_{[ji]k\dots }=-A_{(ij)k\dots }B_{[ij]k\dots }=0.$

Важливий антисиметричний тензор у фізиці — тензор електромагнітного поля F.

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