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Index notation and tensors

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

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

    pasmith

    User Avatar
    Homework Helper

    [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?

    [tex]
    \nabla \cdot (\nabla u) = \nabla (\nabla \cdot u)
    [/tex]
     
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