anonanon 152k1212 gold badges233233 silver badges398398 bronze badges $endgroup$ 4 $begingroup$ That's great many thanks a bunch! I also posted the same concern below: math.stackexchange.com/queries/2135276/… that You may have some insight on? $endgroup$
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ε i j ε i n = δ j n displaystyle varepsilon _ ij varepsilon ^ in = delta _ j ^ n
Samples of Levi-Civita Attributes in 4 Proportions include the symmetric and antisymmetric character of the tensor, its relation for the metric tensor, and its transformation properties underneath coordinate transformations.
From this behavior, we can certainly deduce what $epsilon^ ij4k $ will be with out going through the computations Yet again: it is just the results of exchanging the final two indices of $epsilon^ ijk4 $,
Sure, there remain many open issues and areas of investigation related to Levi-Civita Homes in four Proportions.
This tensor shouldn't be confused Together with the tensor density field pointed out higher than. The presentation On this portion closely follows Carroll 2004.
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The next are samples of the overall identity earlier mentioned specialized to Minkowski space (Along with the adverse signal arising within the odd range of negatives while in the signature from the metric tensor in possibly sign Conference):
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On the pseudo-Riemannian manifold, 1 may well define a coordinate-invariant covariant tensor subject whose coordinate representation agrees While using the Levi-Civita symbol where ever the coordinate method is these types of that The idea of your tangent Area is orthonormal with regard towards the metric and matches a particular orientation.
E α β γ δ = g α ζ g β η g γ θ g δ ι E ζ η θ ι . displaystyle E^ alpha beta gamma delta =g^ alpha zeta g^ beta eta g^ gamma theta g^ delta iota E_ zeta eta theta iota ,.
Do you see how this could have an effect on the value of $epsilon^ alphabetagammadelta epsilon_ alpha'beta'gamma'delta' $? $endgroup$