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How can I prove the equations without KroneckerDelta or LeviCivitaTensor?

These 729 equations:

81 equationsenter image description here

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    $\begingroup$ What do you mean when you say you want to prove a relation between KroneckerDelta and LeviCivitaTensor without usingKroneckerDelta and LeviCivitaTensor? $\endgroup$ – Jens Feb 20 '17 at 4:55
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One way is simply brute-force.

Here is some setup:

ϵ[i_, j_, k_] := Evaluate[LeviCivitaTensor[3]][[i, j, k]];
δ = KroneckerDelta;
i = Range[1, 3];

The variable i is just a list from 1 to 3, which is what each index can run over. We can use Outer to build up the difference of the LHS and RHS and then use AllTrue to check that everybody is 0.

AllTrue[# == 0 &]@*Flatten@Outer[
δ[#1, #4] δ[#2, #5] δ[#3, #6]
+ δ[#2, #4] δ[#3, #5] δ[#1, #6]
+ δ[#3, #4] δ[#1, #5] δ[#2, #6]
- δ[#2, #4] δ[#1, #5] δ[#3, #6]
- δ[#1, #4] δ[#3, #5] δ[#2, #6]
- δ[#3, #4] δ[#2, #5] δ[#1, #6]
- ϵ[#1, #2, #3] ϵ[#4, #5, #6]
&, i, i, i, i, i, i]

A similar exercise for the 81 is

AllTrue[# == 0 &]@*Flatten@Outer[
  δ[#1, #3] δ[#2, #4] - δ[#1, #4] δ[#2, #3]
  - Sum[ϵ[#1, #2, k] ϵ[#3, #4, k], {k, i}]
  &, i, i, i, i]

There are other identities too, such as

AllTrue[# == 0 &]@*Flatten@Outer[
    2 δ[#1, #2] - Sum[ϵ[#1, j, k] ϵ[#2, j, k], {j, i}, {k, i}] &, i, i]

and

Sum[ϵ[ii, j, k] ϵ[ii, j, k], {ii, i}, {j, i}, {k, i}]
| improve this answer | |
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  • $\begingroup$ thanks.Can I use the same way to proof another 9 equations [Epsilon][[i,j,k]][Epsilon][[l,j,k]]=2[Delta] and the last one equation [Epsilon][[i,j,k]][Epsilon][[i,j,k]]=6? $\endgroup$ – Jone Will Feb 21 '17 at 20:39
  • $\begingroup$ Updated with the other identities. $\endgroup$ – evanb Feb 22 '17 at 8:05

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