[Vitagliano 2017]
Let $E \rightarrow M$ a fiber bundle and $W \rightarrow M$ a smooth vector bundle. An $W$-valued $k$-th order differential operator on $E$ is a bundle map
$$ F:J^k E \rightarrow W $$($J^kE$ is the jet bundle).
Every differential operator determines a map $\Delta_F:\Gamma(E)\rightarrow \Gamma(W)$
$$ \Delta_F(s):=F\circ j^k s $$The zero locus $Z(F)$ of $F$ is the preimage $F^{-1}(0_W)$ being $0_W$ the zero section on $W$.
If we assume some regularity conditions on $F$, $Z(F)$ is a system of DEs. A section of $E$ such that $\Delta_F(s)=0$ is a solution and conversely.
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Author of the notes: Antonio J. Pan-Collantes
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