Once we generate the Levi-Civita symbol as a determinant like we did higher than, the thoroughly antisymmetric residence that it possess will become evident: swapping any two indices corresponds to interchanging their corresponding rows in the matrix resulting from which the determinant, that's the Levi-Civita itself, alterations sign.
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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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The phrase "n-dimensional Levi-Civita symbol" refers to The point that the quantity of indices within the image n matches the dimensionality on the vector House in query, which may be Euclidean or non-Euclidean, one example is, R three displaystyle mathbb R ^ three
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then still left facet is zero, and proper facet is likewise zero considering the fact that two of its rows are equivalent. Likewise for j c = j c + one displaystyle j_ c =j_ c+1
The Levi-Civita tensor is used in the Einstein subject equations to express the curvature of spacetime with regard to the energy and momentum of matter and radiation.
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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 levis4d 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 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 &=-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 ,.conclusion aligned
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