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

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In summary, 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. This definition also applies to a rank 3 totally antisymmetric tensor, with the caveat that a cyclic permutation is considered even/odd depending on whether the number of elements being permuted is odd/even. However, the definition does not take into account the behavior of raising and lowering indices, making the rank 4 tensor a pseudo-tensor and the rank 3 tensor a true tensor. Therefore, while the definition is the same, there are important distinctions between the two.
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ehrenfest
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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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  • #2
Mmmmmm. Yes! Do you think 4 is special?
 
  • #3
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.
 
  • #5
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.
 

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

What is a totally antisymmetric tensor?

A totally antisymmetric tensor is a type of mathematical object that represents a multilinear mapping between vector spaces. It is characterized by having all of its components change sign when any two indices are swapped.

What are some examples of totally antisymmetric tensors?

Some examples of totally antisymmetric tensors include the Levi-Civita tensor, the alternating symbol, and the Hodge dual. These tensors have important applications in fields such as physics, engineering, and geometry.

What are the properties of a totally antisymmetric tensor?

A totally antisymmetric tensor has the properties of being invariant under orthogonal transformations, being zero when any two indices are equal, and having components that change sign when any two indices are swapped.

What is the significance of totally antisymmetric tensors in physics?

Totally antisymmetric tensors play a crucial role in various areas of physics, such as electromagnetism, general relativity, and quantum mechanics. They are used to describe physical quantities that are independent of the choice of coordinate system.

How are totally antisymmetric tensors used in engineering?

In engineering, totally antisymmetric tensors are used in applications such as stress analysis, fluid dynamics, and control systems. They help to describe physical phenomena and make calculations more efficient and accurate.

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