Question

Tensor calculus question, how do I show that the covariant derivative of a covariant vector component is zero? In symbols...

Answer 1

How do I show this:\[\Large \nabla _k Z^k= 0\]

Answer 2

@ivancsc1996

Answer 3

ur teaching this or asking it ?

Answer 4

Nope I'm asking XD also I just realized I wrote it wrong. whoops

Answer 5

I fixed it and added as far as I could solve it. I guess I need to figure out how to make the Christoffel symbol and the covariant component turn into the Kronecker delta? \[\Large \nabla_k Z^i=\frac{\partial Z^i}{\partial Z^k}-\Gamma_{kj}^i Z^j \\ \Large \delta_k^i-\Gamma_{kj}^iZ^j\]

Answer 6

i need definitions of covariant :|

Answer 7

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Answer 8

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Answer 9

This is the problem with tensor notation. You have to be careful. If you mean what you wrote then you are asking for the covariant derivative of the coordinates to be zero which is not true in general. On the other hand, if you asked for the covariant derivative of the contravariant basis, then you can look at page 112 and the section on metrilinic property of Pavel's book

Answer 10

Ah thanks, I guess I was just ahead of the book in his lectures. Right now I just started on chapter 6 and I have to say I'm enjoying it quite a bit, very fascinating. But I couldn't find anything about the metrinillic property after he mentioned it in his lectures and I was trying to show it to myself. Thanks again.

Answer 11

Great man!