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ε i j k ε l m n =
I am using the Einstein notation, so various indices are summed) or they give two 3-indices symbols with unique indicator
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Is it possible for the humanoid to find out English within just a few days, just from observing conversations?
This tensor really should not be puzzled While using the tensor density area outlined higher than. The presentation On this portion closely follows Carroll 2004.
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In any arbitrary curvilinear coordinate system as well as in the absence of the metric about the manifold, the Levi-Civita symbol as described previously mentioned could be thought to be a tensor density field in two various ways.
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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 = − two δ ρ σ α β E α β γ δ E ρ σ θ δ = − δ ρ σ θ this site 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 &=-twodelta _ rho sigma ^ alpha beta E^ alpha beta gamma delta E_ rho sigma theta delta &=-delta _ rho sigma theta ^ alpha beta gamma ,.stop aligned
exactly where gab is definitely the representation on the metric in that coordinate process. We will similarly consider a contravariant Levi-Civita tensor by increasing the indices While using the metric as typical,
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