Find the first variation of the following functionals:

Problem #1: Find the first variation of the following functionals: du 2 dv 2 du dv a) G(u, v,u', v') = cuz + c,v2 + cub+ CA + C's + C6 (10 points) dx dx dx dx b b) I(u) = = N/ - [[(u")2 + 2u'u" + (u')? -2u dx (10 points) a c) I(u, v) =- -[ (u2 + 2uv + 2uv +v. +2fu+2gv)dxdy (10 points) 2 2