I'm looking for information on how to formulate and use the Euler-Lagrange equations for finding extrema in a problem with multiple independent variables. Is there an equivalent of the Beltrami Identity if the integrand of the functional has no explicit dependence on the independent variables? I'm trying to determine the means to find the parametrized surface (x(u,v),y(u,v),z(u,v)) for which the integral of a certain function (of x,y,z, and the first partial derivatives of x,y,z with respect to u and v) is a local extremum, and I've only ever worked with the calculus of variations and Euler-Lagrange equation for problems with a single independent variable. (The physics courses at Caltech where I learned the topic only covered single independent variable problems as all the problems we worked with had time as the independent variable). Any help would be appreciated.

--Kevin