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Mystery (?) operator

  1. Apr 22, 2008 #1

    Mentz114

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    Gold Member

    Has anyone come across this operator ?

    [tex]\frac{\partial^2}{\partial_x\partial_y} + \frac{\partial^2}{\partial_x\partial_z} + \frac{\partial^2}{\partial_z\partial_y}[/tex]

    I've never seen it until it came up in a field theory context. What can it mean ?
     
  2. jcsd
  3. Apr 22, 2008 #2
    Hm, I've never seen it used in any context before. Where did you see it, maybe it's interesting to me, as well.:smile:
     
  4. Apr 22, 2008 #3
    I was just fiddling around a bit and I noticed that the operator above equals

    [tex] \frac{1}{2} \left ( \begin{bmatrix}
    0 &1 &1\\
    1 &0 &1\\
    1 &1 &0
    \end{bmatrix} \nabla \right ) \cdot \nabla[/tex]

    which kind of reminds me of the scalar triple product

    [tex]\frac{1}{2} \nabla \cdot \left ( \begin{bmatrix} 1\\ 1\\ 1\end{bmatrix} \times \nabla \right )[/tex]
     
  5. Apr 23, 2008 #4

    Mentz114

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    The operator is half the divergence of a 3D symmetric curl,

    [tex]\nabla^{i} \cdot ( s_{ijk}\partial^{j}A^{k})[/tex].

    where s is a symmetric permutation operator, which is like the Levi-Civita symbol, but with positive value where the L-C has a negative.( I can't write this in vector notation just now)

    I'm not sure it has any physical interpretation. Given that the divergence of the usual curl is identically zero, I thought this thing might mean something.

    foxjwill, you are right, I noticed also what you say. I haven't come across that weird matrix in other context.

    M
     
    Last edited: Apr 23, 2008
  6. Apr 23, 2008 #5

    Mentz114

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    Gold Member

    foxjwill,

    I've realized that this

    [tex] \frac{1}{2} \left ( \begin{bmatrix}0 &1 &1\\1 &0 &1\\1 &1 &0\end{bmatrix} \nabla \right )[/tex]

    is another way to write the symmetric curl.
     
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