O'Neill's Elementary Differential Geometry contains an argument for the following proposition:
"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 dg is never zero on M. I could not show this, except in particular cases such as $f=ax+by$ ($dg=adx+bd\rho$), $f=xy$ ($dg=\rho dx+xd\rho$) or $f=x^2+(y-1)^2$ ($dg=2xdx+(pdy+qdz)$)
where $\rho=\sqrt{y^2+z^2}$
How to show it generally?
I can show it for a more general case: $y=h(x)$, $f(x,y)=y-h(x)$
$dg=d\rho-h^\prime dx$ where $\rho=\sqrt{y^2+z^2}$