Is It Possible to Independently Vary Kij in Both Methods?

[nenyan]

I got the result that is consist with references in fisrt case. Is there anything wrong in 2nd way?

 

[Orodruin]

Both are wrong. In the first case you are missing a 2 in both terms (still giving the same equation if you put the variation to zero). In the second case you missed a 2 only on the first term.

 

[nenyan]

Thank you. I see.
K^ij=g^img^jnK_mn
I should consider another K_ij here.

 

[Orodruin]

Yes, you cannot vary K_ij independently of K^ij.