Calculus of variations: Lagrange multipliers

Solution 1:

The short answer is yes and yes. As a reference, I would recommend (highly) an inexpensive general math reference that I have found helpful: Mathematical Handbook for Scientists and Engineers. It has practically every applied math theorem and technique you'd want. Note that it is a math reference, not a computational/engineering reference, so it discusses the theoretical aspects. I found answers to your questions on pages 11.6-3 and 11.6-9, respectively. The general Lagrange multiplier $\lambda(x)$ has, as a specific case the constant function $\lambda_{i}$ for an isoparametric constraint of the type you indicated in (2).

Hope that helps.