- #1
littleHilbert
- 56
- 0
Hi all! I've got a short question concerning a minor notational issue about tensor contraction I've run across recently.
Let A be an antisymmetric (0,2)-tensor and S a symmetric (2,0)-tensor.
Then their total contraction is zero: [itex]C_1^1C_2^2\,A \otimes S=0[/itex].
As a proof one simply computes: [itex]A_{ij}S^{ij}=-A_{ji}S^{ji}=-A_{ij}S^{ij}[/itex]
When I first saw this, I was a bit confused about the second equality. Of course, a scalar is a symmetric tensor…but is it not an abuse of notation? I mean this seems to run into conflict with the way one handles components of antisymmetric tensors…as for me, for someone who's just got accustomed to the components manipulation machinery, I was disturbed when I saw this. Am I alone?
This is not a big deal…but are there alternatives to expressing stuff like that? Any comments?
Let A be an antisymmetric (0,2)-tensor and S a symmetric (2,0)-tensor.
Then their total contraction is zero: [itex]C_1^1C_2^2\,A \otimes S=0[/itex].
As a proof one simply computes: [itex]A_{ij}S^{ij}=-A_{ji}S^{ji}=-A_{ij}S^{ij}[/itex]
When I first saw this, I was a bit confused about the second equality. Of course, a scalar is a symmetric tensor…but is it not an abuse of notation? I mean this seems to run into conflict with the way one handles components of antisymmetric tensors…as for me, for someone who's just got accustomed to the components manipulation machinery, I was disturbed when I saw this. Am I alone?
This is not a big deal…but are there alternatives to expressing stuff like that? Any comments?