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Proof of a vectoral differentation identity by levi civita symbol

  1. Mar 2, 2013 #1
    1. The problem statement, all variables and given/known data

    prove,
    ∇x(ψv)=ψ(∇xv)-vx(∇ψ)
    using levi civita symbol and tensor notations

    2. Relevant equations

    εijkεimnjnδkmknδjm

    3. The attempt at a solution


    i tried for nth component

    εnjk (d/dxjklm ψl vm

    εknjεklm (d/dxj) ψl vm

    using εijkεimnjnδkmknδjm

    i got,

    (d/dxjn vj - (d/dxjj vn

    But, i can't go further. I think only one simple step is left to show it is equal to the right hand side of the given identity. But how?
     
  2. jcsd
  3. Mar 2, 2013 #2

    dx

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    Hi advphys,

    That identity is not applicable here. Why do you have two levi-civita symbols? And ψ is a scalar, not a vector.

    (∇ x ψv)i = εijk(∂/∂xj)ψvk
     
  4. Mar 2, 2013 #3
    Because i may not know any other identity. :D

    I thought for (ψv)k term i may have one more levi civita symbol.
     
  5. Mar 2, 2013 #4

    dx

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    Just evaluate the derivative using the product rule, and remember that (∇ψ)i = ∂ψ/∂xi
     
  6. Mar 2, 2013 #5

    dx

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    ψv is a vector with components ψvi, so the cross product ∇ x ψv is simply the vector with components εijk(∂/∂xj)(ψvk)
     
  7. Mar 2, 2013 #6
    Oh, yes. Definitely.
    Thanks a lot, i got that.
     
    Last edited: Mar 2, 2013
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