# Meaning of totally antisymmetric tensor

by homology
Tags: antisymmetric, meaning, tensor, totally
 P: 307 Simple question I am confused on. If I have a tensor $M^{\alpha\beta\gamma}$ 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
 PF Patron HW Helper Sci Advisor P: 4,755 What is your definition of a tensor, and of a totally antisymmetric tensor (resp. totally symmetric tensor)?
 P: 590 Take a look at the Levi-Civita symbol http://en.wikipedia.org/wiki/Levi-Civita_symbol
P: 307

## 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 $\partial_{\alpha}F_{\beta\gamma}$. I know that $\partial_{$\alpha}F_{\beta\gamma$}=0$ and I then INCORRECTLY assumed that $\partial_{\alpha}F_{\beta\gamma}$ was totally symmetric which was leading me into errors....argh....noob mistake.

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