It is an operator acting on functions with domain in the jet bundle $J^n$ (differential functions). In the case of $p=q=1$ is given by
$$ \sum_{j=0}^n \left(-D_x\right)^j \dfrac{\partial}{\partial u_{j}} $$being $D_x$ the total derivative operator.
For the motivation for this definition see variational derivative#Some facts.
In the general case is defined as
$$ \mathbf{E}_\alpha=\sum_J(-D)_J \frac{\partial}{\partial u_J^\alpha} $$(see @olver86 page 246).
Euler operator gives 0 when applied to a function $L$ coming from a total divergence. This is shown in @olver86. Is part of a complex, the variational bicomplex with the operators: $\mbox{Div}$ and Helmholtz. See my own question in MO.
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Author of the notes: Antonio J. Pan-Collantes
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