The discussion revolves around the nature of the object Q defined by the components Qij=AiBj, where A and B are vectors. Participants are exploring whether this object qualifies as a tensor of rank 2, particularly focusing on its transformation properties under rotations.
Some participants express a growing understanding of the concept, indicating that they are considering the implications of transformation rules for tensors. There is a recognition that proving something is a tensor involves applying transformations, although the discussion has not reached a consensus on the proof itself.
Participants are working within the constraints of a homework assignment, which may limit the information they can reference or the methods they can use to prove their points.
Well, how do the vectors ##A## and ##B## transform under rotations?flintbox said:Homework Statement
Suppose A and B are vectors. Show that the object Q with nine components Qij=AiBj is a tensor of rank 2.
Homework Equations
A tensor transforms under rotations (R) as a vector:
Tij'=RinRjmTnm
The Attempt at a Solution
I wanted to just create the matrix, but I don't know how to prove that this is also a tensor.
flintbox said:Thanks a lot!
I think I understand it now:
$$A_i' B_j' = R_{in}A_n R_{jm}A_m$$
$$A_i' B_j' = R_{in} R_{jm} (A_n A_m)$$
$$A_i' B_j' = R_{in} R_{jm} Q_{nm}$$
$$A_i' B_j' = Q'_{nm}$$
So we for proving something is a tensor, we just apply some transformations to it, right?