I have to find an example of a surface of revolution excluding a sphere and a cone.
Is $\sigma(x,y)=(\cos x, 5, x^2+y^2)$ such an example?
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I also have to find an example of a surface the image of which is not the graph of a smooth function $z=f(x,y)$.
Is $\sigma(x,y)=(3\sqrt{3}, 10\sqrt{y}, 0)$ such an example?
On
For the first part:
The surface of revolution is parameterized as :
$(x, y, z) = ( u \cos(v), u \sin(v), z= f(u) )$
u can be taken as any other smooth function g(u) also.
As you know for a cone
$$ u = z \tan (\alpha) ;\, z= u \cot (\alpha), $$
and sphere
$$ u = \sqrt{ 1-z^2} ;\, z = \sqrt{ 1-u^2}, $$
it can be used for sweeping any meridian like a paraboloid of revolution here..
$$ u = \sqrt{ 4 f z } ; z = u^2/(4f) $$
Is second question, like
$$ z = \frac{x^2+y^2}{x^2-y^2} ? $$ .. did not understand question well.
Take any curve $z=f(r)$. Then $z=f(\sqrt{x^2+y^2})$ will be a surface of revolution. The other question is unclear...