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Differential geometry

  1. Jul 19, 2007 #1
    1. The problem statement, all variables and given/known data
    On R^3 with the usual coordinates (x,y,z), consider the pairs of vector fields X,Y given below. For each pair, determine if there is a function f:R^3-->R with non-vanishing derivative df satisfying Xf=Yf=0, and either find such a function or prove that there is none.
    (a) X=(e^x)d/dx - ((e^x)z + 2y)d/dz, Y=(e^x)d/dy - (2y)d/dz
    (b) X=(e^x)d/dx - ((e^x)z + 2x)d/dz, Y=(e^x)d/dy - (2y)d/dz
    2. Relevant equations

    Could you help me start this problem.

    3. The attempt at a solution

    Sorry I don't know how to start this problem.
  2. jcsd
  3. Jul 19, 2007 #2


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    You'll want to start by looking up Frobenius integrability conditions.
  4. Jul 24, 2007 #3
    More hint or idea.

    OK I looked at Frobenius integrability condition and still have no clue.
    How can I use the integrability condition??
  5. Jul 24, 2007 #4


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    What is the integrability condition stated in terms of vector fields? You are talking about stuff I haven't looked at for a long time, but isn't the existence of a solution to the PDE's corresponding to these vector fields related to the commutator of the vector fields?
  6. Jul 24, 2007 #5
    Frobenius' Theorem

    Yes. A distribution D={E(x)} on a manifold M is integrable iff for all vector fiels V,W on M with V(x),W(x) in E(x) for all x in M, [v,w](x) in E(x) for all x in M. But how can I relate this with the problem I asked?
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