Lie Derivative of a Vector Field

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.

Re: Lie Derivative of a Vector Field

The stars should be on the bottom and represent pushforwards