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Homework Help: Patches and Surfaces (Differential Geometry)

  1. Feb 23, 2010 #1
    I'm completely confused with patches, which were introduced to us very briefly (we were just given pictures in class). I am using the textbook Elementary Differential Geometry by O'Neill which I can't read for the life of me. I'm here with a simple question and a somewhat harder one.

    1. The problem statement, all variables and given/known data
    Is the following mapping x:R^2 to R^3 a patch?

    x(u,v)=(u, uv, v)?


    2. Relevant equations

    For a mapping to be a patch, it must be one-to-one (injective) and regular (smooth).

    3. The attempt at a solution

    I understand how to show that it is regular; for any arbitrary direction, either the directional derivative of the x component or the directional derivative of the y component is non-zero. Now, I don't know how to prove that it is injective. The book gives a hint: x is one-to-one iff x(u,v) = x(u_1, v_1) implies (u,v)=(u_1,v_1).

    So my attempt was to just let x(u_1,v_1) = (u_1, u_1v_1, v_1) so that
    (u_1, u_1v_1, v_1)=(u, uv, v). Is this the correct way of going about it? I feel like I didn't show anything.

    Can someone also point me toward a better book or online notes where I can try to understand some of this material?

    Thank you, any help or suggestions will be appreciated.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 23, 2010 #2
    Suppose that x(u,v)=x(w,z). Then (u,uv,v)=(w,wz,z), and this implies (u,v)=(w,z).
     
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