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Student Solution Manual for Mathematical Methods for Physics and Engineering, Third Edition by M. P. Hobson, K. F. Riley

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22

Calculus of variations

image

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 image stationary with respect to small variations is given by ρ(z) = k + z2/(4k), where k = [a ± (a2c2)1/2]/2.

image

The surface element lying between z and z + dz is given by

and the integral to be made stationary is

The integrand F

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