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Math Help - reparametrization HELP!!!

  1. #1
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    Question reparametrization HELP!!!

    PLEASE CAN ANYONE HELP ME?? I am completely stuck on this geometry!! If there is a geometry-genious out there: THANK YOU!!
    **i have attached the original question sheet!**

    (sorry, had no maths symbols, so note: E stands for "element of". And f is supposed to be SIGMA, and g is supposed to be TAU)

    Let U = {(u, v) E R2 | 0 < v < u < 1} and
    f(u, v) = (u, u − v, u^2 + v)
    for (u, v) E U.

    a. Show that f is a regular plane, and determine a basis for the tangent space Tpf, for every p E U.
    Prove, that there does not exist a p E U for which e1 E Tpf . Where e1 is the standard basis vector (1,0,0) in R3.

    b. Let
    W =]0, 1[]0, 1[= {(s, t) E R2 | 0 < s < 1, 0 < t < 1} and

    g(s, t) = (s, s − st, s^2 + st)

    for (s, t) E W. Show that g is a reparametrization of f.



    c. Let

    p(s, t) = (s2 + t, t, st^2)
    for (s, t) EV , where V is an open interval, which includes the pointq = (1/2, 0 ). Show that e1 ETqp and show how p cannot be a reparametrization of f.
    Attached Files Attached Files
    Last edited by andrux; February 23rd 2009 at 06:45 AM. Reason: attach question sheet
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  2. #2
    Super Member Rebesques's Avatar
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    Ok... some hints.


    a) Show || f_u\times f_v||\neq 0 for all (u,v). A basis for T_pf is \{f_u(p),f_v(p)\}.

    To show that e_1\notin T_pf \ \forall p, express e_1 as a linear combination of the basis vectors and show this cannot be.


    b) Compute \frac{\partial(u,v)}{\partial(s,t)}(s,t) and show it is nonzero for all (s,t).


    c) Check that (P_s\times P_t)(q)=(0,0,1), so that the tangent space T_qP contains e_1.
    Last edited by Rebesques; June 9th 2009 at 05:54 AM. Reason: being drowsy
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