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Levi-Civita proofs for divergence of curls, etc

  1. Sep 17, 2008 #1
    I've also posted this in the Math forum as it is math as well.

    ---

    I want to know if I'm on the right track here. I'm asked to prove the following.

    a) [tex]\nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla \times \vec{A}) - \vec{A} \cdot (\nabla \times \vec{B})[/tex]
    b) [tex]\nabla \times (f \vec{A}) = f(\nabla \times \vec{A}) - \vec{A} \times (\nabla f)[/tex] (where f is a scalar function)

    And want (read: need to, due to a professor's insistence) to prove these using Levi-Civita notation. I've used the following for reference:
    http://www.uoguelph.ca/~thopman/246/indicial.pdf and http://folk.uio.no/patricg/teaching/a112/levi-civita/

    Here's my attempts - I need to see if I have this notation down correctly...

    a) [tex]\nabla \cdot (\vec{A} \times \vec{B}) [/tex]

    = [tex]\partial_i \hat{u}_i \cdot \epsilon_{jkl} \vec{A}_j \vec{B}_k \hat{u}_l[/tex]

    = [tex]\partial_i \vec{A}_j \vec{B}_k \hat{u}_i \cdot \hat{u}_l \epsilon_{jkl}[/tex]

    Now I thought it'd be wise to use the identity that [tex]\hat{u}_i \cdot \hat{u}_l = \delta_{il}[/tex].

    = [tex]\partial_i \vec{A}_j \vec{B}_k \delta_{il} \epsilon_{jkl}[/tex]

    In which we make i = l (and the [tex]\delta_{il}[/tex] goes to 1).

    = [tex]\partial_i \vec{A}_j \vec{B}_k \epsilon_{jki} [/tex]

    Then using 'scalar derivative product rules' we get two terms. Now, here's where I get a little mixed up. I'm wondering if we rearrange the terms and then modify the epsilon to go in order the the terms.

    = [tex] \vec{B}_k \partial_i \vec{A}_j \epsilon_{kij} + \vec{A}_j \partial_i \vec{B}_k \epsilon_{jik}[/tex]

    Now since the first epsilon is 'even' it remains positive, the other epsilon is 'odd' so that term becomes negative and we end up with the required result.

    = [tex] \vec{B} (\nabla \times \vec{A}) - \vec{A} (\nabla \times \vec{B}) [/tex]

    b) [tex]\nabla \times (f \vec{A}) [/tex] (where f is a scalar function)

    = [tex] \partial_i f \vec{A}_j \hat{u}_k \epsilon_{ijk}[/tex]

    = [tex]f \partial_i \vec{A}_j \hat{u}_k \epsilon_{ijk}+ \vec{A}_j \partial_i f \hat{u}_k \epsilon_{jik}[/tex]

    Once again, the first epsilon is the positive ('even') while the other is negative ('odd').

    = [tex] f (\nabla \times \vec{A}) - \vec{A}(\nabla f)[/tex]

    Man, my hands hurt from all that tex work :P Been awhile for me.
    Since my teacher refuses to tell me if this is the correct method (he's only willing to show the concepts, and while I can appreciate that I don't want my mark to go to hell), can anyone help me out?
     
  2. jcsd
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