Prove the following theorem concerning the local (finite subsidiary) constraint in a calculus of variations problem.
Among all smooth curves whose ennd points lie on two given surfaces z = f(x,y) and z = g(x,y), find the curve for which the the functional J(x,y) = integral{limits z1 to z2} {integrand F(z, x, y, dx/dz, dy/dz)} dz. Please write the differential equations and the boundary conditions. Extra credit of 10 points for suggesting a physically meaningful problem that has this form. You need to then indicate what the functions f(x,y) g(x,y) are.