Show that the cylinder { in } is a regular surface, and find parametrizations whose coordinate neighborhoods cover it.
Please help.
I won't do it for you but..
Parameterize it using cylindrical coordinates into some parameterization S.
To show it is regular you can
-Show the first partial derivative of S are independant.
or
-Show that the cross product of the first partial derivatives of S is non zero
there are many other ways to show it is regular.