Proving the contracted epsilon identity

In summary, the contracted epsilon identity is defined as ε_{ijk}ε_{lmn} = δ_{il}(δ_{jm}δ_{kn} - δ_{jn}δ_{km}) - δ_{im}(δ_{jl}δ_{kn} - δ_{jn}δ_{kl}) + δ_{in}(δ_{jl}δ_{km} - δ_{jm}δ_{kl}) . When contracting the first index in the product, the expected result is δ_{jm}δ_{kn} - δ_{jn}δ_{km} . However, due to an error in the calculation, an extra minus sign was obtained. This was due to the fact that \delta_{ii} =
  • #1
demonelite123
219
0
proving the "contracted epsilon" identity

in the wikipedia page for the Levi Civita symbol, they have a definition of the product of 2 permutation symbols as: [itex] ε_{ijk}ε_{lmn} = δ_{il}(δ_{jm}δ_{kn} - δ_{jn}δ_{km}) - δ_{im}(δ_{jl}δ_{kn} - δ_{jn}δ_{kl}) + δ_{in}(δ_{jl}δ_{km} - δ_{jm}δ_{kl}) [/itex] and by contracting the first index in the product (so that i = l) it should be the case that i get [itex] δ_{jm}δ_{kn} - δ_{jn}δ_{km} [/itex].

however, when i actually replace all the i's with l's i get: [itex] ε_{ijk}ε_{imn} = δ_{ii}(δ_{jm}δ_{kn} - δ_{jn}δ_{km}) - δ_{lm}(δ_{jl}δ_{kn} - δ_{jn}δ_{kl}) + δ_{ln}(δ_{jl}δ_{km} - δ_{jm}δ_{kl}) [/itex] and then using the fact that δ will be 0 unless both of its indices match, i get [itex] ε_{ijk}ε_{imn} = (δ_{jm}δ_{kn} - δ_{jn}δ_{km}) - (δ_{jm}δ_{kn} - δ_{jn}δ_{km}) + (δ_{jn}δ_{km} - δ_{jm}δ_{kn}) [/itex],

but this turns out to be [itex] - (δ_{jm}δ_{kn} - δ_{jn}δ_{km}) [/itex] which is the negative of the answer that I expected. did i do something wrong? I don't know why i picked up an extra minus sign.
 
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  • #2


Your problem is that [itex]\delta_{ii} = 3[/itex], not 1. The first term should have a leading factor of 3.
 
  • #3


Muphrid said:
Your problem is that [itex]\delta_{ii} = 3[/itex], not 1. The first term should have a leading factor of 3.

doh! thank you for pointing that out.
 

1. What is the contracted epsilon identity?

The contracted epsilon identity is a mathematical identity that relates the Kronecker delta and the Levi-Civita symbol. It is defined as:
δijεijk = δki

2. Why is it called the "contracted" epsilon identity?

The term "contracted" refers to the operation of summing over repeated indices. In the contracted epsilon identity, the index k is repeated, hence the name "contracted" epsilon identity.

3. What is the significance of the contracted epsilon identity?

The contracted epsilon identity is significant in many areas of physics, particularly in the study of vector calculus, differential geometry, and quantum mechanics. It is used to simplify and solve equations involving the Levi-Civita symbol, which represents the cross product in three-dimensional vector calculus.

4. How is the contracted epsilon identity proven?

The contracted epsilon identity can be proven using the properties of the Kronecker delta and the Levi-Civita symbol, as well as the definition of the cross product. By substituting the definitions and manipulating the indices, the contracted epsilon identity can be derived.

5. What are some applications of the contracted epsilon identity?

The contracted epsilon identity is used in various fields of physics and mathematics, such as electromagnetism, fluid mechanics, and general relativity. It is also used in computer graphics and engineering for vector computations and transformations.

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