Understanding the Levi-Civita Identity: Simplifying the Notation

In summary, when \epsilon_{mni}a_{n}(\epsilon_{ijk}b_j c_{k}) is rearranged, it becomes \epsilon_{imn}\epsilon_{ijk}a_{n}b_j c_{k}, which is the same expression with the indices of \epsilon_{mni} changed to \epsilon_{imn} due to the commutativity of real numbers. This allows for the use of the identity \epsilon_{mni} = \epsilon_{imn} in the summation.
  • #1
cozmo
2
0
Can somebody show me how

[itex]\epsilon_{mni}a_{n}(\epsilon_{ijk}b_j c_{k})[/itex]

Turns in to

[itex]\epsilon_{imn}\epsilon_{ijk}a_{n}b_j c_{k}[/itex]


Something about the first [itex]\epsilon[/itex] I'm not seeing here when the terms are moved around.
 
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  • #2
The tensor changes sign when you transpose two indices, right? A cyclic permutation is an even number of transpositions, so [itex]\epsilon_{mni} = \epsilon_{nim} = \epsilon_{imn}[/itex].
 
  • #3
When [itex]\epsilon_{ijk}[/itex] and [itex]a_{n}[/itex] change places the [itex]\epsilon_{mni}[/itex] changes to a cyclic permutation that is still positive and [itex]\epsilon_{mni} =\epsilon_{imn}=\epsilon_{nim} [/itex] but each one of these will give a different final answer.

I don't see how [itex]\epsilon_{mni} [/itex] turns to [itex]\epsilon_{imn} [/itex] when [itex]\epsilon_{ijk} [/itex] doesn't change.

This is from a problem proving the A X (B X C) = (A*C)B-(A*B)C identity.
 
  • #4
You can swap [itex]a_n[/itex] and [itex]\epsilon_{ijk}[/itex] because real numbers commute. Swapping them has nothing to do with reordering the indices of [itex]\epsilon_{mni}[/itex].

[itex]\epsilon_{mni} = \epsilon_{imn}[/itex] for all i, m, and n, so you can simply replace [itex]\epsilon_{mni}[/itex] with [itex]\epsilon_{imn}[/itex] in the summation. There's no relabeling of indices going on if that's what you think is happening.
 

1. What is the Levi-Civita identity?

The Levi-Civita identity is a mathematical formula that describes the relationship between the partial derivatives of a vector field. It is also known as the cyclic identity or the alternating tensor identity.

2. Why is the Levi-Civita identity important?

The Levi-Civita identity is important in many areas of mathematics and physics, particularly in the fields of vector calculus, differential geometry, and electromagnetism. It is used to simplify and solve complex equations and has applications in areas such as fluid dynamics, general relativity, and quantum mechanics.

3. How is the Levi-Civita identity derived?

The Levi-Civita identity is derived from the properties of the Levi-Civita symbol, which is a mathematical symbol used to represent the orientation of a coordinate system. By manipulating the properties of the symbol, the Levi-Civita identity can be derived.

4. Can the Levi-Civita identity be generalized to higher dimensions?

Yes, the Levi-Civita identity can be generalized to any number of dimensions. In three dimensions, it is written as a triple sum, but in higher dimensions, it becomes a multi-index sum. The generalization of the identity is useful in higher-dimensional calculus and differential geometry.

5. What are some common applications of the Levi-Civita identity?

Some common applications of the Levi-Civita identity include solving vector calculus problems such as curl and divergence, calculating the surface integral of a vector field, determining the curvature of a surface in differential geometry, and solving Maxwell's equations in electromagnetism.

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