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.



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> **Question:** in which *global* (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?
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In which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?
Igor gave a coordinate independent definition of quasilinear *equations* which the Euler-Lagrange equations satisfy. Still missing is a definition of quasilinear differential *operator*, which the Euler-Lagrange operator satisfies. 


You already observed that



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> $M\times\_M V^\circledast E$ is generally *not* a vector or affine bundle over $M$.
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hence the definition of quasilinear operator found in Bryant, Chern et. al. on p. 397 (which agrees with the definition found in Krasil'shchik, Lychagin, Vinogradov, [Geometry of jet spaces and nonlinear partial differential equations](http://cds.cern.ch/record/335544) p.160) cannot be applied directly, since they assume vector bundles over $M$. But $V^\circledast E$ is a vector bundle over $E$, and the fix might be to expand the definition of nonlinear DO: 


**Definition:** Let $V\to E$ be a vector bundle over $E$ and $\psi:J^k E \to V$ a morphism of fiber bundles *over $E$* (not neccessarily linear, since we don't assume a linear structure on $E\to M$), then the map $\psi \circ j^k$ is called a *nonlinear DO*. Call such an operator *quasilinear* if $\psi$ is affine, when restricted to any fiber of $J^k E \to J^{k-1} E$.


I haven't checked, but suspect that the E-L-operators satisfies this definition of quasilinear DO (with $V=V^\circledast E$). Observe that the zero set of a quasilinear $\psi$ is a quasilinear PDE $\mathcal{E}\subset J^k E$, as in Igors answer. Observe also that a nonlinear DO in the more restrictive sense of: a map of fiber bundles $\rho: J^k E \to W$ over $M$ is a special case of the previous definition by pulling back $W$ to $E$.


Note also that such a generalization of DO is unavoidable in cases like minimal surfaces, where we don't have a fiber bundle structure on the ambient space $E$. (In fact, in the case of jets of submanifolds we probably need to allow vector bundles $V$ over $J^1 E$ instead of over $E$, since the vertical bundle $VE$ doesn't exist and is replaced by a "normal" bundle.)


