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.
$V_u\pi $ is the vector space tangent to the fiber $\pi^{-1}(x)$ of the bundle $\pi : E\to M$ going through $u$. In the book local coordinates $(x',u')$ provide $T_u E$ with the splitting $V_u \pi + H_u E$, where (horizontal) subspace is identified with $T_u M$. We may think of the horizontal space as the tangent to locally constant sections $\phi: M\to E$ going through the point $u$, i.e., $\phi(x')\equiv u$. Now, taking vertical differential of a section $\phi$ equals the projection of the ordinary differential $d\phi$ to $V_u E$ along this $H_u$. Jet manifold $J^1_π$ has projections on both $E$ and $M$ which makes it bundle, it is called associated since its structure group is defined by the structure group of the initial bundle $\pi$, see (3.5).