Editing Euler–Lagrange equation (section)

===Single function of several variables with single derivative===
A multi-dimensional generalization comes from considering a function on n variables. If <math>\Omega</math> is some surface, then

: <math> 
   I[f] = \int_{\Omega} \mathcal{L}(x_1, \dots , x_n, f, f_{1}, \dots , f_{n})\, \mathrm{d}\mathbf{x}\,\! ~;~~
      f_{j} := \cfrac{\partial f}{\partial x_j}
</math>

is extremized only if ''f'' satisfies the [[partial differential equation]]

: <math> \frac{\partial \mathcal{L}}{\partial f} - \sum_{j=1}^{n} \frac{\partial}{\partial x_j}\left(\frac{\partial \mathcal{L}}{\partial f_{j}}\right) = 0. </math>

When ''n'' = 2 and functional <math>\mathcal I</math> is the [[energy functional]], this leads to the soap-film [[minimal surface]] problem.