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Homework Help: Help with tensor notation and curl

  1. Oct 3, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that [itex]\nabla \times (a \cdot \nabla a) = a\cdot\nabla(\nabla\times a) + (\nabla \cdot a(\nabla \times a) - (\nabla \times a)\cdot \nabla a[/itex]

    2. Relevant equations

    [tex]\nabla \times (\nabla \phi) = 0[/tex]
    [tex]\nabla \cdot (\nabla \times a) = 0[/tex]

    3. The attempt at a solution

    I started with breaking the LHS into two components. Observing that:
    [tex]a\times(\nabla\times a) = 0.5 \nabla(a\cdot a) - a\cdot\nabla a[/tex]

    Taking the cross product of both sides:
    [tex]\nabla\times a\times(\nabla\times a) = 0.5\nabla\times\nabla(a\cdot a) - \nabla\times(a\cdot \nabla a)[/tex]

    Recognizing that the term on the right corresponds to our initial equation.
    [tex]\nabla \times (a \cdot \nabla a) = -\nabla\times a\times(\nabla\times a)[/tex]

    Unfortunately, I am sort of stuck here. One way that I have thought to go about it is by calling the LHS:
    [tex]-\nabla\times a\times(\nabla\times a) = -\nabla\times c[/tex]
    where c = a\times(\nabla\times a)
    I am confused on how to expand the above out using levi civita. I know that:
    [tex](\nabla \times c)_i = \epsilon_{ijk} \frac{dc_k}{dx_j} [/tex]
    But substituting in for c isn't making sense to me.

    Sorry for the very rough attempt at a solution. I only started doing vector calc a week ago.
    Last edited: Oct 3, 2011
  2. jcsd
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