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The dynamics of a field theory is specified by an equation of motion, a partial differential equation for such sections. Since differential equations are equations among all the derivatives of such sections, we consider the spaces that these form: the jet bundle $J^\infty_\Sigma E $ is the bundle over $\Sigma$ whose fiber over a point $\sigma \in \Sigma$ is the space of sections of $E$ over the infinitesimal neighbourhood $\mathbb{D}_\sigma$ of that point:
Therefore every section $\phi$ of $E$ yields a section $j^\infty(\phi)$ of the jet bundle, given by $\phi$ and all its higher order derivatives.
Accordingly, for $E, F$ any two smooth bundles over $\Sigma$, then a bundle map
encodes a (non-linear) differential operator $D_f : \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(F)$ by sending any section $\phi$ of $E$ to the section $f \circ j^\infty(\phi)$ of $F$.https://arxiv.org/abs/1601.05956