*22*

*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* + *z*2/(4*k*), *where* *k* = [*a* ± (*a*^{2} – *c*^{2})^{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*

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