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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?