Meaning of totally antisymmetric tensor

[homology]

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

 

[quasar987]

What is your definition of a tensor, and of a totally antisymmetric tensor (resp. totally symmetric tensor)?

 

[M Quack]

Take a look at the Levi-Civita symbol

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

 

[homology]

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.

 

[blue_raver22]

A totally antisymmetric tensor is a type of tensor that has a specific property where it changes sign when any two indices are exchanged. This means that if we have a tensor M^{\alpha\beta\gamma}, and we swap the positions of any two indices, the resulting tensor will have a negative sign compared to the original tensor. In other words, the value of the tensor is dependent on the order of its indices.

As for the follow-up question, a totally symmetric tensor, on the other hand, does not change sign under any pair exchange of indices. This means that the value of the tensor remains the same regardless of the order of its indices.

These properties are important in the study of tensors as they allow us to categorize and understand their behavior and properties. I hope this explanation helps to clarify your confusion.