ΩRm\Omega\in\mathbb R^m is a closed region. The variable f:ΩRp\mathbf f:\Omega\rightarrow\mathbb R^p is an nn differentiable function with fixed boundary conditions on Ω\partial\Omega. The function L\mathcal L is real-valued and has continuous first partial derivatives, and the 00th to nnth partial derivatives of f\mathbf f and the independent variable xΩ\mathbf x\in\Omega will be arguments of L\mathcal L. Define a functional

IfΩL()dV, I\coloneqq\mathbf f\mapsto\int_\Omega\mathcal L\left(\cdots\right)\mathrm dV,

where dV\mathrm dV is the volume element in Ω\Omega. Then the extremal of II satisfies a set of PDEs with respect to f\mathbf f. The set of PDEs consists of pp equations, the iith of which is

j=0nμPj,m(1)jμL(μfi)=0,\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, (1)(1)

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

μlenμkxμk. \partial_\mu\coloneqq\frac{\partial^{\operatorname{len}\mu}}{\prod_k\partial x_{\mu_k}}.

Equation 1 is the Generalization of Euler–Lagrange equation.