Solving the Mystery of the Operator: $\partial^2$

In summary, the conversation discusses a mathematical operator that some have not seen before and is reminiscent of a scalar triple product. The operator is half the divergence of a 3D symmetric curl and may not have a clear physical interpretation. It is also noted that the operator can be written in another form, similar to a symmetric curl.
  • #1
Mentz114
5,432
292
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 ?
 
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  • #2
Mentz114 said:
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:
 
  • #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]
 
  • #4
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:
  • #5
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.
 

1. What is the purpose of solving the mystery of the Operator: $\partial^2$?

The purpose of solving the mystery of the Operator: $\partial^2$ is to better understand its role in mathematical equations and its applications in various fields such as physics, engineering, and economics. By understanding this operator, we can gain insights into the behavior of complex systems and make more accurate predictions.

2. What is the definition of the Operator: $\partial^2$?

The Operator: $\partial^2$ is a mathematical symbol that represents the second derivative of a function with respect to one or more variables. It is commonly used in differential equations to describe rates of change and curvature.

3. How is the Operator: $\partial^2$ different from the regular derivative?

The Operator: $\partial^2$ is different from the regular derivative because it represents the rate of change of the rate of change (or curvature) of a function, whereas the regular derivative only represents the rate of change. In other words, the operator $\partial^2$ takes the derivative of the derivative.

4. What are some real-world applications of the Operator: $\partial^2$?

The Operator: $\partial^2$ has various applications in different fields. In physics, it is used to describe the behavior of waves and particles. In engineering, it is used to model and analyze the dynamics of systems. In economics, it is used to study the behavior of markets and economic trends.

5. How do scientists solve the mystery of the Operator: $\partial^2$?

Scientists solve the mystery of the Operator: $\partial^2$ by using mathematical analysis, numerical methods, and computer simulations. They also conduct experiments and observations to test their theories and models. Collaborative efforts and peer review also play a crucial role in solving this mystery.

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