describe the surfaces r=constant theta=constant and a =constant in the cylindrical coordinate system

Question

A question arises in my Multivariable calculus book that appears as follows:

Describe the surfaces

$r=$constant,

$\theta$ = constant,

$z$ = constant

in the cylindrical coordinate system.

I am unsure they mean to consider each one separately or together. If r is constant we have a circle, and if z is constant we have a flat 2-dimensional plane translated z units, not sure about theta. Any thoughts or answers appreciated.

There is a similar question concerning spherical co-ordinates: if

$\rho$ is constant

$\theta$ is constant

$\phi$ is constant

Answer 1

From the text it seems that they wish you to describe the 3 surfaces, each having one variable as a constant.

Regarding your question, you are close but not quite correct.
If $\rho=constant$ then you get a cylinder. You don't get a circle since your circle can be in different heights according to the $z$ axis.
if $\theta=constant$ then you get half a plane excluding the z axis.
if $z=constant$ then you get a plane parallel to $(x,y,0)$.

Elaboration on $\theta$: keep in mind that there is no top, no bottom and it goes on forever since: $\matrix{{z\in(-\infty,\infty)}\\{\rho\in(0,\infty)}&{}&}$

Spherical Coordinates:
now, I'm going to answer in $(\rho,\theta,\phi)$ meaning $\phi$ is the opening from the $z$ axis and $\theta$ is in plane $xy$.

if $\rho=constant$ then you get a sphere.

if $\theta=constant$ you will once again get a plane, the same as last time. Imagine that you have a constant angle at which you can look up and down since $\phi\in(-\frac{\pi}{2},\frac{\pi}{2})$ and there is no limit to your range since $\rho>0$.

if $\phi=constant$ you get a circle that's increasing in it's radius as you climb up the z axis. Try to imagine it, you have an increasing radius and a circle in every radius. You get a cone. Elaboration on $\phi$: