Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Extracting a matrix from a curl operation

  1. Jun 3, 2013 #1
    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!
     
  2. jcsd
  3. Jun 3, 2013 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    The equations are linear, I think you can just take individual components of [A] and see if it works.
     
  4. Jun 3, 2013 #3
    Hi mfb,

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

    Thanks
     
  5. Jun 3, 2013 #4

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    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).
     
  6. Jun 3, 2013 #5
    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
     
  7. Jun 3, 2013 #6

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    The same result? They should not give the same result. Different results are fine.
     
  8. Jun 3, 2013 #7
    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
     
  9. Jun 3, 2013 #8

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    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).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook