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Need assistance(Gussian curvature and differentiable vector fields)

  1. May 26, 2009 #1
    Need urgent assistance(Gussian curvature and differentiable vector fields)

    Hi I have a very difficult problem where I know some of the dots but can't connect them :(

    So therefore I hope that there is someone who can assist me (hopefully :))

    1. The problem statement, all variables and given/known data

    Let S be a surface with orientation N. Let [tex]V \subset S[/tex] be an open set in S and let [tex]f: V\subset S \rightarrow \mathbb{R}[/tex] be any nowhere zero differentiable function in V. Let [tex]v_1[/tex] and [tex]v_2[/tex] be two differentiable (tangent) vector fields in V such that at each point of V, [tex]v_1[/tex] and [tex]v_2[/tex] and that [tex]v_1 \land v_2 = N[/tex]

    Then prove that [tex]K = \frac{<d(fN)(V_1) \land d(fN)(V_2), fN>}{f^3}[/tex]

    p.s. there is also a question two but since this is so difficult I live that out for the time being hoping we can get to that later.

    2. Relevant equations


    3. The attempt at a solution

    Here is what I know

    Since S has the orientation N that according to do Carmo Geometry book means that it can be covered with a neighbourhood N.

    From what I get is that

    dN(v1) = cv1 + dv2 and dN(v2) = ev1 + fv2 but how do Carmo goes from that the above is a mystery to me. So therefore I hope there is someone who would help me understand what I am missing ?

    Cheers
    Louisa
     
    Last edited: May 26, 2009
  2. jcsd
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