After we produce the Levi-Civita symbol as being a determinant like we did above, the thoroughly antisymmetric assets that it possess will become apparent: swapping any two indices corresponds to interchanging their corresponding rows in the matrix as a result of which the determinant, that's the Levi-Civita by itself, changes indication.
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= δ i l ( δ j m δ k n − δ j n δ k m ) − δ i m ( δ j l δ k n − δ j n δ k l ) + δ i n ( δ j l δ k m − δ j m δ k l ) .
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These Attributes are important for understanding the habits of your tensor in several coordinate techniques.
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First off english will not be my mother tongue sorry. I choose to question if you can enable me with a number of the Attributes in the levi-civita image.
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Utilizing the funds pi notation Π for standard multiplication of quantities, an specific expression to the symbol is:[citation necessary]
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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 = here − 6 δ ρ α E α β γ δ E ρ σ γ δ = − 2 δ ρ σ α β E α β γ δ E ρ σ θ δ = − δ ρ σ θ α β γ . displaystyle start off 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 &=-sixdelta _ 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 ,.stop 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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