Differential operator

[Vitagliano 2017]
Let $E\to M$ a fiber fiber bundle and $W\to M$ a smooth vector fiber bundle. An $W$-valued $k$-th order differential operator on $E$ is a bundle map

($J^{k}E$ is the jet bundle).

Every differential operator determines a map ${\mathrm{\Delta }}_F:\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(W)$

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 ${\mathrm{\Delta }}_F(s)=0$ is a solution and conversely.