$\Omega\in\mathbb R^m$ 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 \begin{equation} I:=\mathbf f\mapsto\int_\Omega\mathcal L\left(\cdots\right)\mathrm dV, \end{equation} 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 fi\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 \begin{equation} \partial_\mu:=\frac{\partial^{\operatorname{len}\mu}}{\prodk\partial x{\mu_k}}. \end{equation} Equation \ref{ret} is the Generalization of Euler–Lagrange equation.
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$\Omega\in\mathbb R^m$ 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 \begin{equation} I:=\mathbf f\mapsto\int_\Omega\mathcal L\left(\cdots\right)\mathrm dV, \end{equation} 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 fi\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 \begin{equation} \partial_\mu:=\frac{\partial^{\operatorname{len}\mu}}{\prodk\partial x{\mu_k}}. \end{equation} Equation \ref{ret} is the Generalization of Euler–Lagrange equation.