Why does the first order necessary condition for constrained optimization require linear independence of the gradients of the equality constraints at the local minimum point?
Liberzon's "Calculus of Variations..." states that it is required but doesn't explain why (see section 1.2.2, p. 11 in print or p. 9 in this free online edition: http://liberzon.csl.illinois.edu/teaching/cvoc.pdf).