Levi-Civita Tensor: Index Interchange Identity

Thread starter AxiomOfChoice
Start date Feb 28, 2009
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Index Levi-civita Properties Tensor

In summary, the Levi-Civita tensor is a rank-2 tensor used in vector calculus and differential geometry to encode information about the orientation of a vector space. It has the index interchange identity, also known as the cyclic property, which allows for the manipulation of indices in calculations. This identity is significant because it simplifies calculations and highlights the antisymmetric nature of the tensor. The tensor also has the properties of orthogonality and completeness.

Feb 28, 2009

AxiomOfChoice

Does the following identity hold?:

[tex]
\epsilon_{ijk} a_j b_k = -\epsilon_{ijk} a_k b_j
[/tex]

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Feb 28, 2009

cristo

Staff Emeritus

Science Advisor

Yes.

Feb 28, 2009

confinement

The answer is yes; clearly we have:
[tex]
\epsilon_{ijk} a_j b_k = -\epsilon_{ikj} a_j b_k
[/tex]

from which we can obtain your identity by relabeling the dummy indices j and k.

1. What is the Levi-Civita tensor?

The Levi-Civita tensor, also known as the permutation tensor, is a mathematical object used in vector calculus and differential geometry. It is a rank-2 tensor that encodes information about the orientation of a vector space.

2. What is the index interchange identity for the Levi-Civita tensor?

The index interchange identity, also known as the cyclic property, states that if any two indices of the Levi-Civita tensor are interchanged, the sign of the tensor changes. This can be written as ε_ijk = -ε_jik = -ε_kji.

3. How is the index interchange identity used in calculations?

The index interchange identity is used in vector calculus and differential geometry to simplify calculations involving the Levi-Civita tensor. It allows for the manipulation of the indices to obtain new expressions that may be easier to work with.

4. What is the significance of the index interchange identity?

The index interchange identity is significant because it allows for the simplification of calculations involving the Levi-Civita tensor. It also highlights the antisymmetric nature of the tensor, where changing the order of the indices changes the sign of the tensor.

5. Are there any other properties of the Levi-Civita tensor?

Yes, the Levi-Civita tensor also satisfies the property of orthogonality, which states that when all three indices are the same, the tensor is equal to 1. It also has the property of completeness, where the sum of all possible permutations of the indices is equal to 0 unless the indices are in a cyclic order, in which case it is equal to 1.