Extracting a matrix from a curl operation

[KrayzBlu]

Hello,

I would like to know if it is possible (and the solution, if known, please!) to extract a 3x3 matrix [A] from a curl operation. Specifically, if B is a 3x1 (column) vector,

∇x([A]B) = [C](∇xB)

What is the value of tensor [C]? Would [C] be a 3x3 matrix as well, or a different rank tensor? Can I express [C] in terms of [A]?

Thanks!

 

[mfb]

The equations are linear, I think you can just take individual components of [A] and see if it works.

 

[KrayzBlu]

Hi mfb,

I've tried that, and unfortunately come up with no solution. I'm wondering of there's some sort of mathemagical trick I've never heard of :)

Thanks

 

[mfb]

What did you get as attempt for
A=
1 0 0
0 0 0
0 0 0

and
A=
0 0 0
1 0 0
0 0 0
?

If both give some corresponding C, it works (based on linearity and symmetry), otherwise it does not work for general matrices A (trivial).

 

[KrayzBlu]

If both give some corresponding C, it works (based on linearity and symmetry), otherwise it does not work for general matrices A (trivial).

They don't give the same result, so I guess there is no general solution. Am I correct in interpreting that this also means that any variation of A cannot be solved for? (For example, if A were diagonal).

Thanks

 

[mfb]

The same result? They should not give the same result. Different results are fine.

 

[KrayzBlu]

Sorry, to clarify, I meant that inserting your suggested [A] matrices both do not give a valid solution for [C]. I'm concluding that if these simple matrices can't be solved for, then it is correct that there is no solution.

Thanks

 

[mfb]

Okay, I checked it, and there is no solution. It does not work for general matrices A, and I am not sure if there are any solutions apart from the trivial one (A=a*identity matrix).