Parametric representation of paraboloid cylinder

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Homework Help Overview

The discussion revolves around the parametric representation of a paraboloid cylinder defined by the equation z = y^3. Participants are exploring how to express this surface parametrically using variables.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are attempting to define the parametric representation r(u, v) for the surface, with some suggesting to set x as one of the parameters. Questions are raised about the existence of a formula for r(u, v) and the conditions under which it can be determined.

Discussion Status

The discussion is active, with participants sharing their attempts to define the parametric equations. Some guidance has been provided regarding the use of variables and the potential need for creative approaches when standard methods do not apply. There is an ongoing exploration of different interpretations of the problem.

Contextual Notes

Participants are considering the flexibility of the variable x and the implications of defining z in terms of y. There is an acknowledgment that not all surfaces can be represented by straightforward functions, which may require alternative methods or coordinate systems.

geft
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The equation is z = y^3. I know how to do normal planes and spheres, but I don't know what to set for r(u,v) when it comes to paraboloid cylinders.
 
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geft said:
The equation is z = y^3. I know how to do normal planes and spheres, but I don't know what to set for r(u,v) when it comes to paraboloid cylinders.

Since x can be anything, make it one of your variables (either u, or v).
 
x = u
y = v
z = v^3

r(u, v) = [u, v, v^3]?

Is there a formula for the r(u, v)?
 
geft said:
x = u
y = v
z = v^3

r(u, v) = [u, v, v^3]?

Is there a formula for the r(u, v)?

If you're asking if there is a formulaic method for determining what r(u,v) should be, then the answer is "sorta". In the case of something along the lines of z(x,y) then you let u and v be x and y, and then just have the function as your z parameter. However you don't always get things defined by functions like that, in which case you need to get a little creative. Either re-arranging things, or even jumping coordinate systems to make stuff easier.
 

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