00:01
Now here in this question, you ask to show that, it said asymmetrical two index tensor, tig, furnishes a vector representation of s -o -3.
00:13
First of, to correct this question, is not the symmetric tensor.
00:17
Instead, it's the anti -symmetric tensor that furnished the vector representation of s -o -3, which is the group of three -dimensional rotations.
00:27
So the correct question is, show that the antithmetric tensor tig furnishes vector representation, right? so free, so i'm going to assume this tig is antithmetric.
00:40
So for antithmetric tensor, you know, the diagonal parts must all be zero, right? zero, zero, and you would have t, let's say, um, t, uh, so you have the two here, you have this here, you are somewhere here, you have somewhere here, you have somewhere here, right? so suppose this is x, let me put it y and z, i'll range them like this, x, y, and z, then this will be txy, tx, z, and this will be y, z, right? and of course, this will be minus t, y, z, and there's minus txd, and minus txy, so the three independent components, txy, tx, y, are actually form a vector representation.
01:25
To show this, actually, we just need to see that, you know, txy, you know, the tig transform exactly in the same way as x, i, x, j, right? x, i, x, j means x, y, z, right? so they transform exact same.
01:43
So in other words, you can, the transformation of txy, t, y, z, t, d, x, they transform in exact same way as xy, y, z, d, x.
01:56
Now, it's very clear that these three, form a vector representation.
02:01
Actually, you can rearrange the three into three components of the following.
02:05
So i can rewrite them as, i can rearrange this as yz minus x and x and x minus yz minus yz...