Meaning of totally antisymmetric tensor


by homology
Tags: antisymmetric, meaning, tensor, totally
homology
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#1
Apr1-12, 08:54 AM
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Simple question I am confused on. If I have a tensor [latex]M^{\alpha\beta\gamma}[/latex] that is totally antisymmetric in its indices then is it the case that M changes sign under the exchange of any two indices? And as a followup, a totally symmetric tensor has no sign changes on any pair exchange of indices?

Thanks,

Kevin
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quasar987
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Apr1-12, 09:09 AM
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What is your definition of a tensor, and of a totally antisymmetric tensor (resp. totally symmetric tensor)?
M Quack
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Apr1-12, 09:16 AM
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Take a look at the Levi-Civita symbol

http://en.wikipedia.org/wiki/Levi-Civita_symbol

homology
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Apr1-12, 09:41 AM
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Meaning of totally antisymmetric tensor


Damn I think I figured out my problem. I was going off Carroll's definition/discussion of (anti-) symmetry (Spacetime and Geometry) which implies that an exchange of a pair of indices in a totally antisymmetric tensor yields a sign change. This is fine I realize now, what is not fine is the following.

I was looking at [latex]\partial_{\alpha}F_{\beta\gamma}[/latex]. I know that [latex]\partial_{\[\alpha}F_{\beta\gamma\]}=0[/latex] and I then INCORRECTLY assumed that [latex]\partial_{\alpha}F_{\beta\gamma}[/latex] was totally symmetric which was leading me into errors....argh....noob mistake.


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