Question

Having so much hard time solving this question...

Original Question is

"Show that the cylinder {(x,y,z) E R^3; x^2 + y^2 = 1} is a regular surface, and find parametrizations whose coordinate neighborhoods cover it"

Okay, I found that cylinder is a regular surface, so now I'm left with solving "find parametrizations whose coordinate neighborhoods cover it"

So I set up the prametrization as g(u,v) = (cosu,sinu,v) where (u,v) E U = (0,2pi) X R.

I showed that g is diffble (checked condition 1)

I showed that one of Jacobian determinants is not 0 (since in U, either one of sinu and cosu is not equal to zero)

I'm now left with solving homeomorphism, which I need to show that g has an inverse and it is continuous.

Other way was showing that g inverse is a restriction of some continuous function, so g inverse is continuous.

How can I show this way?

Is my proof of other conditions are correct?

I'm really having a hard time solving the condition 2...

Answer 1