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[QUOTE="nigelscott, post: 5888824, member: 389949"] I am trying to figure how to get 1. from 2. and vice versa where the e's are bases for the vector space and θ's are bases for the dual vector space. 1. T = T[SUP]μ[/SUP][SUP]ν[/SUP][SUB]σ[/SUB][SUB]ρ[/SUB](e[SUB]μ[/SUB] ⊗ e[SUB]ν[/SUB] ⊗ θ[SUP]σ[/SUP] ⊗ θ[SUP]ρ[/SUP]) 2. T[SUP]μ[/SUP][SUP]ν[/SUP][SUB]σ[/SUB][SUB]ρ[/SUB] = T(θ[SUP]μ[/SUP],θ[SUP]ν[/SUP],e[SUB]σ[/SUB],e[SUB]ρ[/SUB]) My attempt is as follows: 2. into 1. gives T = T(θ[SUP]μ[/SUP],θ[SUP]ν[/SUP],e[SUB]σ[/SUB],e[SUB]ρ[/SUB])(e[SUB]μ[/SUB] ⊗ e[SUB]ν[/SUB] ⊗ θ[SUP]σ[/SUP] ⊗ θ[SUP]ρ[/SUP]) Now if I assume that (θ[SUP]μ[/SUP],θ[SUP]ν[/SUP],e[SUB]σ[/SUB],e[SUB]ρ[/SUB]) Ξ (θ[SUP]μ[/SUP] ⊗ θ[SUP]ν[/SUP] ⊗ e[SUB]σ[/SUB] ⊗ e[SUB]ρ[/SUB]) this becomes: T = T(θ[SUP]μ[/SUP] ⊗ θ[SUP]ν[/SUP] ⊗ e[SUB]σ[/SUB] ⊗ e[SUB]ρ[/SUB])(e[SUB]μ[/SUB] ⊗ e[SUB]ν[/SUB] ⊗ e[SUP]σ[/SUP] ⊗ θ[SUP]ρ[/SUP]) = θ[SUP]μ[/SUP]e[SUB]μ[/SUB] ⊗ θ[SUP]ν[/SUP]e[SUB]ν[/SUB] ⊗ e[SUB]σ[/SUB]θ[SUP]σ[/SUP] ⊗ e[SUB]ρ[/SUB]θ[SUP]ρ[/SUP] Now using θ[SUP]ν[/SUP]e[SUB]μ[/SUB] = δ[SUP]ν[/SUP][SUB]μ[/SUB] this becomes: T = T(I ⊗ I ⊗ I ⊗ I) So T = T This seems to work but I'm not sure if this is the correct way to do it. I'm shaky on the tensor product stuff and my interpretation of T(_,_,_,_). Does this look right? [/QUOTE]
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