Let $\displaystyle X,Y$ be vector fields on a smooth manifold $\displaystyle M$ and let $\displaystyle \phi_t$ be the flow of $\displaystyle X$.

I was trying to write the lie derivative of $\displaystyle Y$ w.r.t. $\displaystyle X$ in local co-ordinates, so I have that

$\displaystyle \phi_{-t}^* \left( \frac{\partial}{\partial x_j} \Big|_{\phi_t(p)} \right) = \displaystyle\sum_{i,j} \frac{\partial \phi_i}{\partial x_j} (-t, \phi_t (p)) \frac{\partial}{\partial x_j} \Big|_p$.

I got this because I know that

$\displaystyle \phi_t^* \left( \frac{\partial}{\partial x_j} \Big|_p \right) = \displaystyle\sum_{i,j} \frac{\partial \phi_ i}{\partial x_j} (t,p) \frac{\partial}{\partial x_j} \Big|_{\phi_t(p)},$

So I just swapped $\displaystyle p$ for $\displaystyle \phi_t(p)$ and $\displaystyle t$ for $\displaystyle -t$.

But I have been told this calculation should give

$\displaystyle \phi_{-t}^* \left( \frac{\partial}{\partial x_j} \Big|_{\phi_t(p)} \right) = \displaystyle\sum_{i,j} \frac{\partial \phi_i}{\partial x_j} (-t, p) \frac{\partial}{\partial x_j} \Big|_p$, and it has to for various proofs to work.

Can anyone tell me why I am wrong? I would appreciate it. Thanks.