What Surfaces Can Be Described Using Cylindrical Coordinates?

[mathmari]

Hey!

I am looking at an exercise that asks to describe the surfaces r=constant, θ=constant and z=constant in the cylindrical coordinate system.

The cylindrical coordinates are $(r, \theta , z)$, that are defined by $x=r \cos \theta , y=r \sin \theta , z=z$

$r=\sqrt{x^2+y^2}, z=z , \theta=\arctan (\frac{y}{x} )$

r=constant=c: $c=\sqrt{x^2+y^2} \Rightarrow x^2+y^2=c^2$ Is this a circle?? (Wondering)

θ=constant=k: What can we say at this case?? (Wondering)

z=constant=m: How can we use this to find the surface?? (Wondering)

 

[MarkFL]

It sounds to me like what is described here is a line parallel to the $z$-axis...

 

[mathmari]

It sounds to me like what is described here is a line parallel to the $z$-axis...

Do we not have to find the surface r=constant, the surface θ=constant and the surface z=constant ?? (Wondering)

Are we looking for only one surface?? (Wondering)

 

[MarkFL]

Are the 3 conditions all to be met simultaneously or are these separate conditions?

 

[mathmari]

Are the 3 conditions all to be met simultaneously or are these separate conditions?

I thought that these 3 conditions are separate since we are asked to describe the surfaces.

 

[MarkFL]

Okay...what do you think $r=k$ where $k$ is a real constant describes?

 

[MarkFL]

I thought that these 3 conditions are separate since we are asked to describe the surfaces.

I have edited the thread title to indicate the fact that surfaces is plural. :D

 

[mathmari]

Okay...what do you think $r=k$ where $k$ is a real constant describes?

Then we have $x^2+y^2=k^2$. If we were in $\mathbb{R}^2$ it would be a circle, right?? (Wondering) but now we are looking in $\mathbb{R}^3$. What is it now?? (Wondering)

I have edited the thread title to indicate the fact that surfaces is plural. :D

OK... :D

 

[Fallen Angel]

Then we have $x^2+y^2=k^2$. If we were in $\mathbb{R}^2$ it would be a circle, right?? (Wondering) but noiw we are looking in $\mathbb{R}^3$. What is it now?? (Wondering)

Right, but you are missing the free variable $z$, so ...

I think it will be easier this time using $x=rcos\theta, \ y=rsin\theta,\ z=z$ rather than the equations you seem to be using in order to see what surface is when one of the variables is constant.

 

[mathmari]

Right, but you are missing the free variable $z$, so ...

I think it will be easier this time using $x=rcos\theta, \ y=rsin\theta,\ z=z$ rather than the equations you seem to be using in order to see what surface is when one of the variables is constant.

Using $x=r \cos \theta$, $y=r \sin \theta$ and $z=z$ and knowing that $r=\text{ constant }=k$, we have that $x=k \cos \theta$ and $y=k \sin \theta$.

How can we see in that way which the surface is?? (Wondering)

 

[MarkFL]

We would have $r=k$ for all $z$...so in every plane $z=\ell$, we would have a circle...what surface does this describe?