The object Q defined by the components Qij=AiBj, where A and B are vectors, is confirmed to be a tensor of rank 2. This conclusion is reached by demonstrating that Q transforms under rotations according to the tensor transformation rule Tij'=RinRjmTnm. The transformation of the vectors A and B under rotations is essential to proving that Q retains its tensorial properties, specifically that Q'_{nm} = A_i' B_j' = R_{in} R_{jm} Q_{nm}.
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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?