Whenever we compose the Levi-Civita image for a determinant like we did previously mentioned, the completely antisymmetric assets that it possess becomes apparent: swapping any two indices corresponds to interchanging their corresponding rows within the matrix as a result of which the determinant, that is the Levi-Civita by itself, modifications indication.
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RexcirusRexcirus 43544 silver badges1515 bronze badges $endgroup$ 2 $begingroup$ Your final expression would not make Substantially feeling: The LHS has free indices $i,j,k$ whereas your RHS has absolutely free indices $i,j,l$. $endgroup$
ε i j ε i n = δ j n displaystyle varepsilon _ ij varepsilon ^ in = delta _ j ^ n
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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 last two indices of $epsilon^ ijk4 $,
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Firstly english just isn't my mom tongue sorry. I desire to ask if you can help me with many of the properties with the levi-civita symbol.
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but observe that If your metric signature incorporates an odd range of destructive eigenvalues q, then the indication of your factors of the tensor differ from your normal Levi-Civita symbol:
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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 = − 6 δ ρ α E α β γ δ E ρ σ γ δ = − two go here δ ρ σ α β E α β γ δ E ρ σ θ δ = − δ ρ σ θ α β γ . displaystyle start out 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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