Levi-Civita symbol

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Levi-Civita symbol

[′lā·vē chē·vē′tä ‚sim·bəl]
(mathematics)
A symbol εi,j ,…,s where i, j, …, s are n indices, each running from 1 to n ; the symbol equals zero if any two indices are identical, and 1 or -1 otherwise, depending on whether i, j,…, s form an even or an odd permutation of 1, 2,…, n.
References in periodicals archive ?
However, Brian Kong and the present author argued in [12] that we arrive at this formula, if we use, in the equation for the area two-form, a Levi-Civita tensor instead of a Levi-Civita symbol as conventionally done in loop quantum gravity community.
where components [[epsilon].sup.123] = [[epsilon].sup.312] of Levi-Civita tensor are equal to 1/[square root of g].
where [omega] = 1/2 [nabla] x u is the vorticity and [sigma] is the dual vector to the antisymmetric constituent of the turbulent stress tensor with components [[sigma].sub.k] = [e.sub.kij][[sigma].sub.ij] ([e.sub.ijk] denotes the Levi-Civita tensor components and the Einstein summation is assumed).
Then the structure constants are Levi-Civita tensor components [f.sub.BCD] [right arrow] [[epsilon].sub.knm], and expressions for potential and intensity (strength) of the gauge field are written as:
We have a remarkable fact: form-invariance of Q-multiplication has as a corollary "covariant constancy" of matrices of spinor transformations of vector Q-units; moreover one notes that proper Q-connexion (contracted in skew indices by Levi-Civita tensor) plays the role of "gauge potential" of some Yang-Mills-type field.