Whenever we compose the Levi-Civita image for a determinant like we did higher than, the fully antisymmetric residence that it possess results in being apparent: swapping any two indices corresponds to interchanging their corresponding rows within the matrix as a consequence of which the determinant, which can be the Levi-Civita itself, changes sign.

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δ i l δ i m δ i n δ j l δ j m δ j n δ k l δ k m δ k n

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Levi-Civita Houses in four dimensions are carefully related to Einstein's principle of typical relativity, which describes the curvature of spacetime while in the existence of huge objects.

From this habits, we can easily deduce what $epsilon^ ij4k $ will be with no dealing with the computations once again: it is simply the result of exchanging the final two indices of $epsilon^ ijk4 $,

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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 &=-twodelta _ rho sigma ^ alpha beta E^ alpha beta gamma delta E_ rho sigma theta delta &=-delta _ rho sigma theta ^ alpha beta gamma ,.close aligned

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 ,.

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