I'm still trying to figure out how to do the Lie derivative of a Jacobian. (c.f. earlier unanswered post).
If I know how how to do Lie derivatives on section of vector bundles, that would be sufficient. In my case the vector bundle $\pi:E\rightarrow X$ is given by the dual of the pullback bundle $E=(\phi^*TY)^*$, where $\phi:X\rightarrow Y$ is a smooth map between smooth manifolds. So let $\alpha$ be a section of that bundle. How do I define $\mathcal L_v\alpha$ ? And it would be helpful also to know it in local coordinates.
The reason why this would help me to define the Lie derivative of the Jacobian is that the Jacobian is a linear map from $TX\otimes (\pi^*TY)^*$ to $\mathbb R$, i.e.
\begin{equation}
(w,\alpha)_x\mapsto \langle D\phi\, w,\alpha\rangle_x\;
\end{equation}
where locally $w=\sum_iw^i(x)\frac{\partial}{\partial x^i}$ and $\alpha=\sum_b\alpha^b(x)\,dy^b$ ($x^i$ a coords on $X$ and $y^b$ are coords on $Y$) and the pairing is $\langle\cdot,\cdot\rangle_x: T_{\phi(x)}Y \times T^*_{\phi(x)}Y\rightarrow \mathbb R$. Then I could define $\big(\mathcal L_vD\phi\big) (w,\alpha)=\mathcal L_v \big(D\phi(w,\alpha)\big)-D\phi(\mathcal L_v w,\alpha)-D\phi(w,\mathcal L_v\alpha)$, knowing $\mathcal L_v\alpha$.
Best Answer
The material mentioned by Yuri in the comment can also be found in "Topics in differential geometry", available here http://www.mat.univie.ac.at/~michor/dgbook.pdf by the same author, section 8.15. It may be a bit simpler than "Natural operations in differential geometry".
I was looking at the material right now. From what I have understood you need to be able to lift the flow of the vector field on the base manifold to the total space of the vector bundle in such a way that
the flow maps fibres to fibres
the action on a given fibre is linear
Given all this the Lie derivative can be defined in the usual why as the $\mathrm{d}/\mathrm{d}t \,|_{t=0}\, \phi^*_t$ where $\phi$ is the lifted flow.
Of course for this to be useful you need to have some preferred/unique way of lifting the flow on the base to the total space.