Show that ${d\over dt} \Phi _t^* \mu= \Phi_t^* L_X\mu$, $\;\Phi _t$ the flow of $X$, $\mu$ a form.

Question

Let $\Phi _t$ the flow of a vector field $X$, and $\mu$ an $m$ form on a manifold $M$. I want to prove the following relation:

$${d\over dt} \Phi _t^* \mu= \Phi_t^* L_X\mu$$

I was looking for an elementary proof of the fact. Thank for your help.

Answer 1

In case you don't get any other answers, this is proved in Proposition 12.36 in my Introduction to Smooth Manifolds (2nd ed.). The proof is only a few lines long.