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Math Help - do Carmo question - differential geometry....

  1. #1
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    do Carmo question - differential geometry....

    Let P = {(x,y,z) : x=y}, and let a function x:U->R^3 be given by x(u,v) = (u+v,u+v, uv) where U = {(u,v) : u>v}

    Is x a parameterization of P?

    I'm really stuck on this question and I feel it is easier than I am making it out to be.

    The answer is Yes, (as it says this in the back of the book) - i'm not really sure how to be rigorous in proving it though.

    I see that in the function x, x = y. so this is a start. I think What the question is asking is that given any (x,y,z) can you find a u and v such that x = y = u+v and uv = z??

    I have got something like x = u + v, z = uv and then have solved for u and v in terms of x and z. Is this all we need to show? I end up getting u and v are derived from the quadratic formula.

    u=\frac{x\pm \sqrt{x^2 - 4z}}{2}

    the same for v...

    this would mean that x^2 needs to be > 4z... does this matter?

    now if u>vwould mean you would need to take the positive sqare root for u and negative for v. is this right?

    If u = v, then it is possible we might run in to problems.

    Please help someone... im so shite at all this
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: do Carmo question - differential geometry....

    The point (0,0,1) belongs to P , however the system \begin{Bmatrix} u+v=0\\uv=1\end{matrix} has no real solutions. Perhaps there is a typo in your book.
    Last edited by FernandoRevilla; September 4th 2011 at 10:02 PM.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Re: do Carmo question - differential geometry....

    Quote Originally Posted by mathswannabe View Post
    Let P = {(x,y,z) : x=y}, and let a function x:U->R^3 be given by x(u,v) = (u+v,u+v, uv) where U = {(u,v) : u>v}

    Is x a parameterization of P?

    I'm really stuck on this question and I feel it is easier than I am making it out to be.

    The answer is Yes, (as it says this in the back of the book) - i'm not really sure how to be rigorous in proving it though.

    I see that in the function x, x = y. so this is a start. I think What the question is asking is that given any (x,y,z) can you find a u and v such that x = y = u+v and uv = z??

    I have got something like x = u + v, z = uv and then have solved for u and v in terms of x and z. Is this all we need to show? I end up getting u and v are derived from the quadratic formula.

    u=\frac{x\pm \sqrt{x^2 - 4z}}{2}

    the same for v...

    this would mean that x^2 needs to be > 4z... does this matter?

    now if u>vwould mean you would need to take the positive sqare root for u and negative for v. is this right?

    If u = v, then it is possible we might run in to problems.

    Please help someone... im so shite at all this
    I feel as though this might be helpful.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: do Carmo question - differential geometry....

    Quote Originally Posted by Drexel28 View Post
    I feel as though this might be helpful.
    So, we need two books.
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