Let be vector fields on a smooth manifold and let be the flow of .
I was trying to write the lie derivative of w.r.t. in local co-ordinates, so I have that
I got this because I know that
So I just swapped for and for .
But I have been told this calculation should give
, and it has to for various proofs to work.
Can anyone tell me why I am wrong? I would appreciate it. Thanks.