Editing Euler–Lagrange equation (section)

===Several functions of several variables with higher derivatives===
If there are ''p'' unknown functions ''f''<sub>i</sub> to be determined that are dependent on ''m'' variables ''x''<sub>1</sub> ... ''x''<sub>m</sub> and if the functional depends on higher derivatives of the ''f''<sub>i</sub> up to ''n''-th order such that
:<math>
   \begin{align}
     I[f_1,\ldots,f_p] & = \int_{\Omega} \mathcal{L}(x_1, \ldots, x_m; f_1,\ldots,f_p; f_{1,1},\ldots,
     f_{p,m}; f_{1,11},\ldots, f_{p,mm};\ldots; f_{p,1\ldots 1}, \ldots, f_{p,m\ldots m})\, \mathrm{d}\mathbf{x} \\
     & \qquad \quad
        f_{i,\mu} := \cfrac{\partial f_i}{\partial x_\mu} \; , \quad
        f_{i,\mu_1\mu_2} := \cfrac{\partial^2 f_i}{\partial x_{\mu_1}\partial x_{\mu_2}} \; , \;\; \dots
   \end{align}
</math>

where <math>\mu_1 \dots \mu_j</math> are indices that span the number of variables, that is they go from 1 to m. Then the Euler–Lagrange equation is

:<math>
    \frac{\partial \mathcal{L}}{\partial f_i} +\sum_{j=1}^n \sum_{\mu_1 \leq \ldots \leq \mu_j} (-1)^j \frac{\partial^j}{\partial x_{\mu_{1}}\dots \partial x_{\mu_{j}}} \left( \frac{\partial \mathcal{L} }{\partial f_{i,\mu_1\dots\mu_j}}\right)=0
 </math>

where the summation over the <math>\mu_1 \dots \mu_j</math> is avoiding counting the same derivative <math> f_{i,\mu_1\mu_2} = f_{i,\mu_2\mu_1}</math> several times, just as in the previous subsection. This can be expressed more compactly as

:<math>
\sum_{j=0}^n \sum_{\mu_1 \leq \ldots \leq \mu_j} (-1)^j \partial_{ \mu_{1}\ldots \mu_{j} }^j \left( \frac{\partial \mathcal{L} }{\partial f_{i,\mu_1\dots\mu_j}}\right)=0
 </math>