- #1

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"Let C be a curve in a plane P and let A be a line that does not meet C. When this *profile curve* C is revolved around the axis A, it sweeps out a surface of revolution M."

For simplicity, he assumes that P is the xy plane and A is the x axis. He says,

"If the profile curve is ##C:f(x,y)=c## we define a function g on ##R^3## by

##g(x,y,z)=f(x,\sqrt{y^2+z^2})##"

He says it is not hard to show, using the chain rule, that the differential dg is never zero on M. I could not show this except in particular cases:

IF ##f=ax+by## then ##dg=adx+bd\rho##

If ##f=xy## then ##dg=\rho dx+xd\rho##

If ##f=x^2+(y-1)^2## then ##dg=2xdx+(pdy+qdz)##

where ##\rho=\sqrt{y^2+z^2}##

How to show it generally?