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Wedge product

  1. Jun 24, 2011 #1
    I have this problem(from Tensor Analysis on Manyfolds by Bishop and Goldberg): prove that
    [itex]e_1^ e_2 + e_3^e_4[/itex] is not decomposable when the dimension of the vector space is greater than 3 and e_i are basis vectors.
    I solved it by mounting a set of 6 equations with 8 unknows and studying the different posibilities cheking that each one is not solvable.
    Is there any nicer way to tackle this problem? if so please let me know
     
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  3. Jun 24, 2011 #2

    tiny-tim

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    hi facenian! :smile:

    (use "\wedge" in latex :wink:)
    you need to prove that it cannot equal [itex]a\wedge b[/itex] where a and b are 1-forms …

    so express a and b in terms of the basis :wink:
     
  4. Jun 24, 2011 #3
    helo tiny-tim, thanks for your prompt response and yes I did what you suggested and it led me to what I explained
     
  5. Jun 24, 2011 #4

    tiny-tim

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    how about [itex]a\wedge (e_1\wedge e_2 + e_3\wedge e_4)[/itex] ? :wink:
     
  6. Jun 28, 2011 #5
    you mean, let [itex]a=\sum_{i<j} x_{ij} e_i\wedge e_j[/itex] and then conclude tha [itex]a[/itex] must be null? Please let me know if that's what you meant and/or if I'm correct
     
  7. Jun 28, 2011 #6

    tiny-tim

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    hi facenian! :smile:

    no, i'm using the same a as before (in a∧b, which you're trying to prove it isn't)

    so let a = ∑i xiei :wink:
     
  8. Jun 28, 2011 #7
    I'm sorry I did not explained it correctly I should have said:

    you mean, let [itex]a=\sum_i x_{i} e_i[/itex] and then conclude tha [itex]a[/itex] must be null because we are left with a linear conbination of basic vectors of the form [itex] \sum x_i e_i\wedge e_j\wedge e_k=0[/itex] .Please let me know if that's what you meant and/or if I'm correct
     
    Last edited: Jun 28, 2011
  9. Jun 28, 2011 #8

    tiny-tim

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    … which has to be 0, because a ∧ (a ∧ b) = 0

    yes :smile:
     
  10. Jun 28, 2011 #9
    thank you very much tiny-tim your method is much better than mine!
     
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