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Homework Help: Index notation and tensors

  1. Feb 24, 2014 #1
    1. The problem statement, all variables and given/known data

    I have a vector v. According to my book, the following is valid:
    \frac{1}{2}\nabla v^2-v\cdot \nabla v = v\times \nabla \times v
    I disagree with this, because the first term on the LHS I can write as (partial differentiation)
    \frac{1}{2}\partial_i v_jv_j = v_j\partial_i v_j
    which is just [itex]v\cdot \nabla v[/itex]. So IMO it should equal 0.

    What is wrong here?
    Last edited: Feb 24, 2014
  2. jcsd
  3. Feb 24, 2014 #2


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    [tex]v_j \partial_i v_j \neq v_j \partial_j v_i = (v \cdot \nabla) v_i[/tex]

    Note that on the left the index on the derivative is i (a free index), but on the right the index on the derivative is j (a dummy index).
  4. Feb 24, 2014 #3
    Thanks. I hope there is omething else you can help me with (you seem to have experience with this kind of notation). Is the following always true for some vector u?

    \nabla \cdot (\nabla u) = \nabla (\nabla \cdot u)
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