In what follows, all manifolds are smooth, Hausdorff, paracompact, connected and oriented, and all maps between any two of them are assumed to be smooth. Let $\pi:E\rightarrow M$ be a fiber bundle over the base manifold $M$, which we assume to have dimension $n$. Given a bundle map $$\mathscr{L}:J^kE\rightarrow\wedge^nT^*M$$ over $M$, where $J^kE$ is (the total space of) the jet bundle of order $k$ of $\pi$ - i.e. $\mathscr{L}$ is a Lagrangian density of order $k$ - we define the *action functional* $S_{\mathscr{L}}:C^\infty_c(M)\times\Gamma(\pi)\rightarrow\mathbb{R}$ associated to $L$ (here $C^\infty_c(M)$ is the space of smooth real-valued functions on $M$ with compact support, and $\Gamma(\pi)=\{\phi\in C^\infty(M,E)\ |\ \pi\circ\phi=\text{id}_M\}$ is the space of smooth sections of $\pi$) as $$S_{\mathscr{L}}(f,\phi)=\int_M f(j^k\phi)^*\mathscr{L}\ .$$ The *Euler-Lagrange operator* $E(\mathscr{L})$ associated to $\mathscr{L}$ is the (usually nonlinear) partial differential operator of order $\leq 2k$ defined by the formula $$\int_M E(\mathscr{L})[\phi](\vec{\phi})=\int_M\left.\frac{\partial}{\partial t}\right|_{t=0}f(j^k\phi_t)^*\mathscr{L}\ ,\quad\vec{\phi}\in\Gamma_c(\phi^*VE\rightarrow M)\ ,f|_{\text{supp}\vec{\phi}}\equiv 1\ ,$$ where $\phi_t=\Phi(t,\cdot)$ for any $\Phi\in C^\infty(I\times M,E)$ with $0\in I\subset\mathbb{R}$ an open interval, $\phi_0=\phi$, $\left.\frac{\partial\Phi}{\partial t}\right|_{t=0}=\vec{\phi}$ and $\phi_t\in\Gamma(\pi)$, $\phi_t|_{M\smallsetminus\text{supp}\vec{\phi}}=\phi|_{M\smallsetminus\text{supp}\vec{\phi}}$ for all $t\in I$. The above definition of $E(\mathscr{L})$ can be shown to be independent of the particular choice of $f$ and $\Phi$ under the above conditions. There are other possible ways to define $E(\mathscr{L})$ in the literature, all leading to the same object.

One can see from the above definition that $E(\mathscr{L})[\phi]\in\Gamma(\phi^*V^\circledast E\rightarrow M)$, where $\pi_{V^\circledast E}:V^\circledast E=\pi^*(\wedge^nT^*M)\otimes V^*E\rightarrow E$ is the so-called *twisted dual* of the *vertical bundle* $VE=\ker T\pi\rightarrow E$ of $E$. Since $$\phi^*V^\circledast E=\{(p,q)\in M\times V^\circledast E\ |\ \phi(p)=\pi_{V^\circledast E}(q)\}\subset M\times_M V^\circledast E$$ for all $\phi\in\Gamma(\pi)$, it is clear that $\phi^*V^\circledast E$ is a sub-bundle of $M\times_M V^\circledast E$ over $M$. As such, we can write $$E(\mathscr{L})=\rho(E(\mathscr{L}))\circ j^{2k}\ ,$$ where $$\rho(E(\mathscr{L})):J^{2k}(E)\rightarrow M\times_M V^\circledast E$$ is a bundle map covering $\text{id}_M$ ($M$ is seen above as a fiber bundle over itself with singleton fibers and $\text{id}_M$ as the projection map).

When one writes a local formula for $E(\mathscr{L})[\phi]$ using a local trivialization of $E$ and a local chart for the typical fiber $Q$ of $\pi$, one sees that $E(\mathscr{L})[\phi]$ is always an *affine* function of the highest-order derivatives of $\phi$ if the order of $E(\mathscr{L})$ happens to be nonzero (which we assume to be the case), so it would make sense to say that $E(\mathscr{L})$ is a "quasi-linear" partial differential operator. However, $M\times_M V^\circledast E$ is generally *not* a vector or affine bundle over $M$ (unless $E$ itself is), despite $\phi^*V^\circledast E$ being a (different) vector bundle over $M$ for each $\phi\in\Gamma(\pi)$, so $E(\mathscr{L})$ does not fit into the usual way to globally define quasi-linear partial differential operators (see e.g. the discussion in Section IX.2, pp. 393ff of the book *Exterior Differential Systems* by R.L. Bryant et al.), which requires the target bundle of the operator to be a vector (or at least an affine) bundle.

Question:in whichglobal(i.e. coordinate-independent) way one may define quasi-linear partial differential operators in order to encompass Euler-Lagrange operators $E(\mathscr{L})$ acting on smooth sections of general fiber bundles?