What is the definition of a rank 3 totally antisymmetric tensor?

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Homework Help Overview

The discussion revolves around the definition of a rank 3 totally antisymmetric tensor, particularly in relation to a rank 4 tensor. Participants are exploring the similarities and differences in definitions and properties of these tensors.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning whether the definition of a rank 3 totally antisymmetric tensor aligns with that of a rank 4 tensor. There is also discussion about the implications of cyclic permutations and the distinction between true tensors and pseudotensors.

Discussion Status

The discussion is ongoing, with participants providing differing perspectives on the completeness of the definitions. Some guidance has been offered regarding the relationship between dimensions and tensor properties, but no consensus has been reached.

Contextual Notes

Participants are considering the implications of raising and lowering indices in the context of tensor definitions. There is also mention of the distinction between rank 3 and rank 4 tensors as true versus pseudotensors.

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Homework Statement


The totally antisymmetric rank 4 tensor is defined as 1 for an even combination of its indices and -1 for an odd combination of its indices and 0 otherwise.

Is a rank 3 totally antisymmetric tensor defined the same way?


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The Attempt at a Solution

 
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Mmmmmm. Yes! Do you think 4 is special?
 
The definition is the same, but remember that a cyclic permuation is even/odd iff the number of elements being permuted is odd/even. This sometimes causes confusion when moving moving from 3 to 4 dimensions.
 
Yes and no, I think the definition here is incomplete. It does not include what happens when you raise and lower an index. The rank 4 anti-symmetric tensor is a psuedotensor, the rank 3 one is a true tensor. So overall no.
 

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