An action integral
$$
S = \int_{\Omega}L(x_1,...,x_n,f,D_{x_1}f, ...,D_{x_n}f) dx_1\cdots dx_n,
$$
can be minimized over a compact set $\Omega$ via the multi-dimensional generalization of Euler-Lagrange equation in the form
$$
\dfrac{\partial L}{\partial f} - \sum_{i=1}^{n} \dfrac{\partial }{\partial x_i} \dfrac{\partial L}{\partial D_{x_i}f} = 0
$$
with $D_{x_i}f := \partial f/\partial x_i$, $f : \mathbb{R}^n\to \mathbb{R}$, and Lagrangian $L(x_1,...,x_n,f,D_{x_1}f, ...,D_{x_n}f)$.
I still couldn't figure out the derivation/proof of this multi-d extension. First of all, could someone point me to a nice derivation/proof of this extended case?
Having said that, I am particularly wondering how we could adopt this equation for arbitrary domains beyond product topology of intervals (i.e., $\Omega = I_1 \times I_2 \times \cdots \times I_n$ for closed intervals $I_i \in \mathbb{R}$). For example, when the domain is a closed n-ball $\Omega = B^n$. Do the same equations hold for any sufficient set of boundary conditions defined on the boundary of the ball $\partial \Omega = S^n$?