Calculus of variations
22.1 A surface of revolution, whose equation in cylindrical polar coordinates is ρ = ρ(z), is bounded by the circles ρ = a, z = ±c (a > c). Show that the function that makes the surface integral stationary with respect to small variations is given by ρ(z) = k + z2/(4k), where k = [a ± (a2 – c2)1/2]/2.
The surface element lying between z and z + dz is given by
and the integral to be made stationary is
The integrand F