What dimension is the surface of a cylindrical shell

[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?

 

[Mark44]

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.

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?

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

 

[kman12]

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?

 

[Mark44]

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.

 

[kman12]

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

 

[Mark44]

Yes.