I'm looking for rigorous discussions on the derivation of the Euler-Lagrange equation for field as it is usually discussed in classical field theory books. More precisely, if the action is given by:

$$S(\phi) = \int \mathcal{L}(\phi, \partial_{x_{i}}\phi) d^{4}\vec{x}$$ where $\vec{x} = (x_{1},x_{2},x_{3},x_{4}=t)\in \mathbb{R}^{4}$ and $\partial_{x_{i}}$ denotes, generically, any of its partial derivatives, then I'm looking for a rigorous derivation of the Euler-Lagrange equation: $$\sum_{i=1}^{4}\frac{\partial}{\partial x_{i}}\bigg{(}\frac{\partial \mathcal{L}}{\partial (\partial_{x_{i}}\phi)}\bigg{)}-\frac{\partial \mathcal{L}}{\partial \phi} = 0 $$