For a vector bundle $(E,\pi, M)$ let $\phi :M\mapsto E$ be a section of $\pi $, $x\in M$ and $u=\phi (x)$. The vertical differential of the section $\phi$ at point $u\in E$ is the map: \begin{equation} d^v_u\phi :T_uE\mapsto \mathcal V_u\pi \end{equation} In coordinates on $E$ $(x^i,u^\alpha)$ we write; \begin{equation} d^v_u\phi =\bigg(du^\alpha -\frac{\partial \phi ^\alpha }{\partial x^i}dx^i\bigg)\otimes \frac{\partial }{\partial u^\alpha} \end{equation} Apparently it is *obvious* from this that $d^v_u\phi$ depends only on the first order jet space $j^1_x\phi$.

What is $\mathcal V_u\pi$ in this case? It is clearly related to the jet manifold $J^1\pi$ whose total space is the product $T^*M\otimes _E\mathcal V\pi$ . But I don't really understand what an *associated vector bundle* is!

References:

- C.M. Campos,
*Geometric Methods in Classical Field Theory and Continuous Media,* pages 24-25.

This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user Janet the Physicist