Discussion Overview
The discussion revolves around the possibility of extracting a 3x3 matrix [A] from a curl operation involving a 3x1 vector B. Participants are exploring the relationship between the curl of the product of a matrix and a vector, specifically questioning the nature and rank of the resulting tensor [C] and its dependence on [A].
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant asks if it is possible to express a tensor [C] in terms of matrix [A] from the equation ∇x([A]B) = [C](∇xB).
- Another participant suggests that the linearity of the equations allows for testing individual components of [A] to find a solution.
- A participant reports unsuccessful attempts to find a solution using specific forms of [A], indicating a lack of a general solution.
- Further discussion reveals that different matrices [A] yield different results for [C], suggesting that a general solution may not exist.
- One participant concludes that if simple matrices do not yield a valid solution for [C], it implies that variations of [A] may also not be solvable.
- Another participant confirms that there is no solution for general matrices [A], proposing that only a trivial solution exists (A=a*identity matrix).
Areas of Agreement / Disagreement
Participants generally agree that there is no solution for general matrices [A], with some suggesting that only trivial solutions may exist. However, there is some debate regarding the implications of testing specific matrices and whether this definitively rules out other forms of [A].
Contextual Notes
The discussion is limited by the assumptions made about the nature of the matrices involved and the specific forms tested. There is also uncertainty regarding the generalizability of the results based on the examples provided.