permutation tensor

permutation tensor

[‚pər·myə′tā·shən ‚ten·sər]
(mathematics)
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For small values, a strain tensor [gamma], an inclination vector [psi] , a p-rotation scalar [OMEGA] , and the permutation tensor E = [E.
micro]v[rho][sigma]] are the components of the completely antisymmetric four-dimensional Levi-Civita permutation tensor density.
ikl] are the components of the completely anti-symmetric three-dimensional Levi-Civita permutation tensor density).
Then the covariant and contravariant components of the totally anti-symmetric permutation tensor are given by
ijk] are the components of the usual permutation tensor density.
In the same manner, we define the four-dimensional permutation tensor as one with components
p]/[square root of det(g) are the covariant and contravariant components of the completely anti-symmetric Levi-Civita permutation tensor, respectively, with the ordinary permutation symbols being given as usual by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[epsilon].
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the covariant and contravariant components of the completely anti-symmetric Levi-Civita permutation tensor, respectively, with the ordinary permutation symbols being given as usual by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
abcd] are the components of the completely anti-symmetric four-dimensional Levi-Civita permutation tensor and [psi] is a vector field normal to a three-dimensional space (hypersurface) [summation](t) defined as the time section ct = [x.