1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Help with tensor notation and curl
  1. Tensor Analysis (Replies: 0)

Loading...