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The Levi-Civita symbol is most often Employed in 3 and 4 Proportions, also to some extent in two Proportions, so these are given right here right before defining the final circumstance. Two Proportions[edit]
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I am utilizing the Einstein notation, so many indices are summed) or they offer two three-indices symbols with various signal
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This tensor should not be confused with the tensor density discipline outlined previously mentioned. The presentation In this particular section carefully follows Carroll 2004.
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Using the capital pi notation Π for ordinary multiplication of figures, an explicit expression to the symbol is:[citation required]
but notice that If your metric signature has an odd range of unfavorable eigenvalues q, then the sign on the components of the tensor vary through the common Levi-Civita image:
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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 = − 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 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 &=-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 ,.close aligned
the place gab may be the illustration on the metric in that coordinate method. We can easily likewise consider a contravariant Levi-Civita tensor by increasing the indices Using the metric as regular,
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