# Homework Help: What dimension is the surface of a cylindrical shell

1. Oct 16, 2009

### kman12

Hello there, i wanted to know what is the difference between a surface in R^2 and a surface in R^3 in an xyz cordiante system. More specifically how do they look like? an example please. Is this different from a 2 dimensional surface and a 3 dimensional surface respectively.

Also what dimension is the surface of a cylindrical shell of height b and radius a in. In an x,y,z co ordinate system?

2. Oct 16, 2009

### Staff: Mentor

Re: surfaces

R2 is the real plane, and represents a flat two-dimensional surface. We don't normally talk about surfaces in R2. The kinds of things that we can represent in R2 are points, lines, curves, and geometric figures that include squares, circles, and so on.
R3 is three-dimensional space, and has one more dimension than R2. An example of a surface in R3 is the outer skin of a ball. A surface can be flat or it can have waves or bumps in it. A surface in space has area, and is therefore two-dimensional, even if it takes three dimensions to represent it.

This surface is two-dimensional. Is that what you're asking? Or are you asking for the area of this surface?

3. Oct 16, 2009

### kman12

Re: surfaces

im just asking how many dimensions the surface of a cylinder has, i guess it would be 2 dimensional since you can make it out of A4 paper which is 2 dimensional. However if you were to draw it on a xyz cordinate system, then each point on that surface would need to be specified by its height width and depth (ie x y z) so i thought it would be a three dimensional surface.
Are all curved surfaces three dimensional? but flat surfaces two dimensional?

4. Oct 16, 2009

### Staff: Mentor

Re: surfaces

The cylinder you described is two-dimensional, but is embedded in a three-dimensional space, which is why you need three coordinates to identify any given point on it. Surfaces by their very nature are two-dimensional, whether flat (planar) or not.

It's similar to the situation in R2, where straight lines are one-dimensional, and so are curves, in all but the most extreme cases (where the curve has so much wiggle to it that it essentially fills up a two-dimensional region). Going up one more dimension, surfaces in space are generally two-dimensional, but I can imagine that there might be some theoretical surfaces that are so convoluted that they fill up a three-dimensional region. I don't remember ever reading about any examples, though.

5. Oct 16, 2009

### kman12

Re: surfaces

right thanks for clearing stuff up, makes good sense now.
So is the paraboloid a three dimensional shape that has a two dimensional surface and is presented in R^3

6. Oct 16, 2009

Re: surfaces

Yes.