Mystery (?) operator

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Mentz114
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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 ?
 

Answers and Replies

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 ?
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:
 
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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]
 
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:
Mentz114
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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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