MHB Finding Cylindrical Points $(r, \theta, z)$ in Range

[evinda]

Hello! (Wave)

How can we find what section of the cylinder $x^2+y^2=1$ corresponds to cylindrical points $(1,\theta,z)$ in the range $\theta$ in $[0,\pi]$ and $z$ in $[ -1,1]$ ?

We have that the cylindrical points are of the form $(r, \theta, z)$ where the following relations hold:

$$x=r \cos{\theta} \\ y=r \sin{\theta} \\ z=z$$

But we cannot use the above relations to use the fact that $\theta \in [0,\pi]$ and $z \in [ -1,1]$, can we? (Thinking)

 

[Evgeny.Makarov]

But we cannot use the above relations to use the fact that $\theta \in [0,\pi]$ and $z \in [ -1,1]$, can we?

Why can't we consider these restrictions with the equations above?

Find the points with coordinates $(1,0,-1),(1,\pi/2,-1),(1,\pi,-1),(1,0,0),(1,\pi/2,0),(1,\pi,0),(1,0,1),(1,\pi/2,1),(1,\pi,1)$. This should give you the idea of what part of the cylinder is considered.

 

[evinda]

Why can't we consider these restrictions with the equations above?

Find the points with coordinates $(1,0,-1),(1,\pi/2,-1),(1,\pi,-1),(1,0,0),(1,\pi/2,0),(1,\pi,0),(1,0,1),(1,\pi/2,1),(1,\pi,1)$. This should give you the idea of what part of the cylinder is considered.

$(x,y,z)$ gets respectively the following values:

$$(1,0,-1) \\ (0,1,-1) \\ (-1,0,-1) \\ (1,0,0) \\ (0,1,0) \\ (-1,0,0) \\ (1,0,1) \\ (0,1,1) \\ (-1,0,1)$$

What do we get from that?

 

[Evgeny.Makarov]

If you plot these points on top of the cylinder, it should give you the idea of what part of the cylinder is considered. You can also consider points where the middle coordinate is $\pi/4$ and $3\pi/4$ in addition to $\pi/2$. But of course it helps to know the geometric meaning of cylindrical coordinates.

In general, I think that one should not ask a question about applying a concept when one does not know the definition of the concept. One should ask for help with the definition instead (if it is hard to find or understand).