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The Levi-Civita symbol

(and odd and even permutations)

So, you've seen something in a mathematical equation that looks something like ε_ijk, and you're just about to panic because you've never seen it before.

Don't panic, it's called the Levi-Civita symbol, and it's defined as follows:

ε_ijk₌

{

-1
1
0

If i, j, k is an odd permutation of 1, 2, 3
If i, j, k is an even permutation of 1, 2, 3
if an index is repeated

where i, j and k stand for the numbers 1, 2 and 3, though not necessarily in that order.

Stop! What is a permutation? What is an even or odd permutation? What is an index?

First of all, the subscript i, j and k are the index numbers, or indices of the Levi-Civita symbol. Basically, ε_ijk_{will be equal to a different value depending on what the i, j and k stand for.}

_{A permutation
of something is just a re-arrangement of it. For example, the numbers
1, 2 and 3 can be arranged as 1,2,3 or 1,3,2 or 2,1,3 or 2,3,1 or 3,1,2
or 3,2,1.}

_{Now, each of these permutations
of 1,2,3 can be obtained by starting with 1,2,3 and swapping two
numbers at a time until you get that permutation. For example, you can
get 3,1,2 from 1,2,3 as follows:}

	Start with:	1, 2, 3
Move 1	swap first and second indices:	2, 1, 3
Move 2	swap first and third indices:	3, 1, 2

_{Here
I did it in two moves. Two is an even number. I took an even number of
moves. Therefore 3,1,2 is an even permutation of 1,2,3.}

_{Mathematically, you can express this using the Levi-Civita symbol as follows:}

ε₃₁₂_{= 1}

What if I take more than 2 moves to get the same result? How do I know for sure that 3,1,2 is an even permutation every time?

It doesn't matter how many moves you take. If you can get a permutation in an even number of moves, that means it's impossible to do it in an odd number of moves.
Likewise, if it takes you an odd number of moves the first time, then you can be sure that permutation can only be obtained from an odd number of moves.

What does it mean that ε_ijk= 0 if an index is repeated?

This just means that if i,j,k is not a permutation of 1,2,3; for example, ε₁₁₂ which has the number 1 repeated and doesn't contain the number 3 at all, then the Levi-Civita symbol will be equal to zero.
This is because some equations include sums over all possible values of i, j and k. This ensures that any non-sensical terms in such an equation are equal to zero and therefore don't count.

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