Generalization of Euler–Lagrange equation

$\Omega\in\mathbb R^m$ is a closed region. The variable $\mathbf f:\Omega\rightarrow\mathbb R^p$ is an $n$ differentiable function with fixed boundary conditions on $\partial\Omega$. The function $\mathcal L$ is real-valued and has continuous first partial derivatives, and the $0$th to $n$th partial derivatives of $\mathbf f$ and the independent variable $\mathbf x\in\Omega$ will be arguments of $\mathcal L$. Define a functional

\[I:=\mathbf f\mapsto\int_\Omega\mathcal L\left(\cdots\right)\mathrm dV,\]

where $\mathrm dV$ is the volume element in $\Omega$. Then the extremal of $I$ satisfies a set of PDEs with respect to $\mathbf f$. The set of PDEs consists of $p$ equations, the $i$th of which is

\[\begin{equation} \sum_{j=0}^n\sum_{\mu\in P_{j,m}}\left(-1\right)^j \partial_\mu\frac{\partial\mathcal L}{\partial\left(\partial_\mu f_i\right)}=0, \label{ret} \end{equation}\]

where $P_{j,m}$ is the set of all (not necessarily strictly) ascending $j$-tuples in $\left\{1,\dots,m\right\}^j$, and

\[\partial_\mu:=\frac{\partial^{\operatorname{len}\mu}}{\prod_k\partial x_{\mu_k}}.\]

Equation \ref{ret} is the Generalization of Euler–Lagrange equation.