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anonanon 152k1212 gold badges233233 silver badges398398 bronze badges $endgroup$ 4 $begingroup$ Which is perfect thanks a bunch! I also posted a similar issue listed here: math.stackexchange.com/issues/2135276/… that You could have some insight on? $endgroup$

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ε i j k ε l m n =

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Samples of Levi-Civita Houses in 4 Proportions involve the symmetric and antisymmetric mother nature with the tensor, its relation to your metric tensor, and its transformation Houses underneath coordinate transformations.

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then remaining aspect is zero, and correct side is also zero because two of its rows are equivalent. Likewise for j c = j c + one displaystyle j_ c =j_ c+one

The Levi-Civita tensor is Employed in the Einstein industry equations to express the curvature of spacetime with regards to the Vitality and momentum of matter and radiation.

In summary, the conversation is with regard to the properties in the Levi-Civita symbol. The 1st section discusses a formula that doesn't manage to match the envisioned remedy, and the 2nd section involves proving An additional formulation. The conversation ends having a last Remedy that satisfies the Homes.

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These contain more knowledge the geometric and Bodily implications of these properties, and their possible generalizations to greater dimensions and other mathematical constructions.

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E α β γ δ E ρ σ μ ν = − g α ζ g β η g γ θ g δ ι δ ρ σ μ ν ζ η θ ι E α β γ δ E ρ σ μ ν = − g α ζ g β η g γ θ g δ ι δ ζ η θ ι ρ σ μ ν E α β γ δ E α β γ δ = − 24 E α β γ δ E ρ β γ δ = − six δ ρ α E α β γ δ E ρ σ γ δ = − two δ ρ σ α β E α β γ δ E ρ σ θ δ = − δ ρ σ θ α β γ . displaystyle start aligned E_ alpha beta gamma delta E_ rho sigma mu nu &=-g_ alpha zeta g_ beta eta g_ gamma theta g_ delta iota delta _ rho sigma mu nu ^ zeta eta theta iota E^ alpha beta gamma delta E^ rho sigma mu nu &=-g^ alpha zeta g^ beta eta g^ gamma theta g^ delta iota delta _ zeta eta theta iota ^ rho sigma mu nu E^ alpha beta gamma delta E_ alpha beta gamma delta &=-24E^ alpha beta gamma delta E_ rho beta gamma delta &=-6delta _ rho ^ alpha E^ alpha beta gamma delta E_ rho sigma gamma delta &=-2delta _ rho sigma ^ alpha beta E^ alpha beta gamma delta E_ rho sigma theta delta &=-delta _ rho sigma theta ^ alpha beta gamma ,.conclude aligned

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Do you see how this may influence the value of $epsilon^ alphabetagammadelta epsilon_ alpha'beta'gamma'delta' $? $endgroup$