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RexcirusRexcirus 43544 silver badges1515 bronze badges $endgroup$ two $begingroup$ Your remaining expression does not make much sense: The LHS has free of charge indices $i,j,k$ whereas your RHS has no cost indices $i,j,l$. $endgroup$
ε i j ε i n = δ j n displaystyle varepsilon _ ij varepsilon ^ in = delta _ j ^ n
These Homes are important for comprehension the behavior on the tensor in different coordinate programs.
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This tensor really should not be confused With all the tensor density discipline stated higher than. The presentation Within this area closely follows Carroll 2004.
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In any arbitrary curvilinear coordinate technique and even in the absence of the metric to the manifold, the Levi-Civita symbol as outlined earlier mentioned may very well be thought of as a tensor density field in two various ways.
but notice that In case the metric signature is made up of an odd variety of destructive eigenvalues q, then the sign of the elements of this tensor vary from the regular 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 learn more 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 &=-twodelta _ rho sigma ^ alpha beta E^ alpha beta gamma delta E_ rho sigma theta delta &=-delta _ rho sigma theta ^ alpha beta gamma ,.conclude aligned
The crucial element defining house in the image is overall antisymmetry from the indices. When any two indices are interchanged, equal or not, the image is negated:
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