Really need urgent help with tensors!!

Question

So say a tensor and vector have components T^α_β^γ, and v^α, respectively in a coordinate system x^μ. There is also another coordinate system x'^μ

 

I want to show that T^α_β^γv^β = T^αβγv_β

 

Please if someone can be so kind to help because I've been stuck on this question for days :(

 

Answer 1

T^α_β^γv^β = T^αηγg_ηβg^βεv_ε  by raising and lowering the indices.

 

 

 

T^α_β^γv^β = T^αηγδ_η^εv_ε by using g_ηβg^βε = δ_η^ε (delta function).

 

 

T^α_β^γv^β = T^αηγv_η by using δ_η^εv_ε = v_η

 

 

Switching out the dummy index η for β, T^α_β^γv^β = T^αβγv_β.