# Math Help - Finding a parametric representation of the surface, (vector calculus) please help me

1. ## Finding a parametric representation of the surface, (vector calculus) please help me

Consider the surface z=thaita (sorry i dont know how to spell it) in cylindrical coordinates shown in figure 3. Find aparametric representation of this surface

I dont even know where to start, please help me out
thanks

2. ## Re: Finding a parametric representation of the surface, (vector calculus) please help

I'll try and help you out, I've never really used cylindrical co-ordinates but I do know that they're defined by three properties; the radial distance from the z-axis, the azimuth (angle to) of the point and height z.
Your equation $\large z=\theta$ would result in a straight line on the z-axis since the distance from the z-axis isn't mentioned. Are you sure the equation is correct as it clearly doesn't correspond with the diagram.

Try it, let $\large r = 0$, $\large \theta =t$ and $\large z = t$ on this grapher, the result is just a straight line.
surf_graph_cylin

3. ## Re: Finding a parametric representation of the surface, (vector calculus) please help

Why $r=0$ ?
Try the plotter with $r=t$, $t=0$ to $5,$ and $\theta = z = s,$ $s = 0$ to $15.$

4. ## Re: Finding a parametric representation of the surface, (vector calculus) please help

Hmmm, I understand what you're saying, but if r isn't explicitly defined surely that means it's either irrelevant hence $r=0$ or it's true for all values of r: $\large r \in \mathbb{R}$, why would the equation $\large z=\theta$ imply that $\large r \in \left [ 0;5 \right ]$?

I'm not criticising you, just brainstorming. =)

I think an appropriate parametric representation would be:

$\large r = t$

$\large z = s$

$\large \theta = s$

where: $\large t \in \mathbb{R}$ and $\large s \in \mathbb{R}$

That does seem a bit silly but it's all I've got at the moment.

5. ## Re: Finding a parametric representation of the surface, (vector calculus) please help

I chose $r = 0\rightarrow 5,$ (and $\theta = z = 0\rightarrow 15),$ for no particular reason other than it gives a reasonable plot (and produces something like the surface shown in the original question) whereas $r = 0\rightarrow\infty$ does not. Just for example really.
If there are no limits specified then I think $\infty$ is correct, afterall if you were asked to plot, in [2D], $y=1$ for example, you would draw, or at least imply, a line of infinite length, not a single point at $(0,1).$

6. ## Re: Finding a parametric representation of the surface, (vector calculus) please help

If the diagram weren't present I'd think the answer I gave was correct but I'm thinking that some information is missing. Given how the diagram looks

$\large r = t$

$\large z = s$

$\large \theta = s$

where: $\large t \in \mathbb{R}$ and $\large s \in \mathbb{R}$

...doesn't seem possible.