Is x^2 + y^2 = 1 ; 0 < z < 1 a manifold

[xoninhas]

Consider the set
M = {(x, y, z ) ? R^3: x^2 + y^2 = 1 ; 0 < z < 1}.

Show that M is a manifold, indicating explicitly the parameterization which the images cover M. Determine M's dimension.

Ok so I just got a layman explanation of what is a manifold... I thought that this way maybe it is better for me to get the bigger picture. this subject follows inverse and implicit functions, so it has probably something to do with that...

I have the solution, I just want to get the understanding of the solution... so step by step is ok

I see this creates a cylinder without base or top, from 0 to 1 (z), and I know a manifold is something that can be drawn like in 2D therefore in R^2...

So with this on my mind what should I do?? Try to convert this into only two variables? trying to see if some variable can be written in function of other 2? What is the system?

I'm sorry so many questions... :S

 

[stapel]

Consider the set M = {(x, y, z ) ? R^3: x^2 + y^2 = 1 ; 0 < z < 1}.

Show that M is a manifold, indicating explicitly the parameterization which the images cover M. Determine M's dimension.

In the text from which you took this exercise, how is a "manifold" defined? What is the definition of the "dimension" of a manifold? Are there any worked examples of what this book means by a "parametrization" whose "image" "covers" a manifold M (and what are the book's definitions of the quote-marked terms)?

Thank you!

Eliz.

 

[xoninhas]

hmmm ok, I wrote manifold cause that was the conversion I got on wikipedia... in portuguese it's written variety (direct english translation I mean), but I think it is meant by manifold (not that my opinion is of any value in this :S )... if it would be variety's dimension would it make more sense?

Damn just translating is hard already

 

[stapel]

I wrote manifold cause that was the conversion I got on wikipedia... in portuguese it's written variety (direct english translation I mean)....

Mathematics runs on definitions. You're asking for explanations of a topic whose definitions you don't know, and of whose names you aren't certain. Unfortunately, it is going to be difficult, if not impossible, to provide the requested lessons, since we can't be quite sure what the question is.

You might want to try consulting with a Portuguese tutoring service for assistance, or else working through a friend who is confortable in all three languages (namely, English, the relevant mathematics, and Portuguese).

Eliz.

 

[xoninhas]

The way I have on my notes, the parameterization is achieved by converting the coordinates to polar coordinates achieving the parameterization:

g1 (?, z ) = (cos ?, sen ?, z ), (?, z ) ? V1 = ]0, 7*pi/4[ × ]0, 1[
g2 (?, z ) = (cos ?, sen ?, z ), (?, z ) ? V2 = ]??, 3?/4[ × ]0, 1[ .

for the dimension and saying that g1 and g2 are parameterizations of M the notes say:

Dg1 (?, z ) = Dg2 (?, z ) = \(\displaystyle \begin{bmatrix}-sin(teta) & 0 \\ cos(teta) & 0 \\ 0 & 1\end{bmatrix}\)

Since both columns are linearly independent g1 and g2 are paremeterizations of M. and g1 U g2 = M. And dimension of the variety (or manifold) is 2.

Maybe you can help me to actually know what is this I'm suppose to study... varieties or manifolds? .... :S

 

[stapel]

Maybe you can help me to actually know what is this I'm suppose to study... varieties or manifolds?

I'm afraid I don't see how we could know this to any degree of certainty...?

Your book (or other resource) is in a language we don't speak. The article on manifolds (provided earlier) apparently did not reflect whatever you're working on. And there appear to be too many different sorts of varieties for me to dare to venture a guess as to which, if any, might possibly relate to your area of study.

In other words, you're asking us to provide you with an accurate translation of your topic into English. But we don't speak the originating language, don't offer translation services, and aren't clear on what the actual topic is. How then to translate it? :shock:

Please consider consulting with a Portuguese-language math-tutoring site for further information. Thank you.

Eliz.