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
} \right) = \displaystyle\sum_{i,j} \frac{\partial \phi_i}{\partial x_j} (-t, \phi_t (p)) \frac{\partial}{\partial x_j} \Big|_p)
.
I got this because I know that
 = \displaystyle\sum_{i,j} \frac{\partial \phi_ i}{\partial x_j} (t,p) \frac{\partial}{\partial x_j} \Big|_{\phi_t(p)},)
So I just swapped 
for )
and 
for 
.
But I have been told this calculation should give
} \right) = \displaystyle\sum_{i,j} \frac{\partial \phi_i}{\partial x_j} (-t, p) \frac{\partial}{\partial x_j} \Big|_p)
, 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
