# Help with parameterization of surface

1. Apr 25, 2012

### Kuma

1. The problem statement, all variables and given/known data

If I have been given a surface x = 12 − y^2 − z^2 between x = 3 and x = 8, oriented by the unit normal which points away from the x–axis.

I want to find an orientation preserving parameterization.
2. Relevant equations

3. The attempt at a solution

I know orientation preserving means that the normal vector is pointing outward. I'm not sure how to apply this to parameterize this surface however.

2. Apr 25, 2012

### LCKurtz

Parameterization and orientation are separate issues. Try cylindrical like coordinates only on y and z instead of x and y.

3. Apr 25, 2012

### Kuma

I figure I can parameterize it no problem but the question literally asks what I said. Find an orientation preserving parameterization. What does that mean?

4. Apr 26, 2012

### LCKurtz

I have seen instances when textbooks say the parameterization itself determines the orientation. For example, if your surface is parameterized as $\vec R =\vec R(u,v)$, then the direction of $\vec R_u \times \vec R_v$ determines the positive orientation of the surface. So, if you parameterize your surface using $r$ and $\theta$, one or the other of $\vec R_r\times \vec R_\theta$ or $\vec R_\theta\times\vec R_r$ will point in the direction that was specified by the problem. If it is the first, then write your parameterization as $\vec R = \vec R(r,\theta)= \ ...$ and if it is the second write it as $\vec R = \vec R(\theta,r)=\ ...$. Personally, I don't care for that notion because, as in your problem, the orientation is given separately. Anyway, that's my best guess what it might mean.

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