Levi-Civita Symbol

Relevant parts to questions...

• It’s the $n$-dimensional generalisation of the Alternating Tensor ${ϵ}_{ijk}$.
• In MTH3008, you mostly meet only the rank-3 form ${ϵ}_{ijk}$ (see Alternating Tensor).
• Identical rules apply: cyclic permutations preserve sign, swaps flip sign, repeated indices give zero.

The Levi-Civita Symbol is the generalised version of the Alternating Tensor, defined for $n$ indices as

$${ϵ}_{i_1{i}_2\dots i_n}=\{\begin{array}{ll}+1& \text{if }(i_1,\dots ,i_n)\text{ is an even permutation of }(1,\dots ,n)\\ -1& \text{if }(i_1,\dots ,i_n)\text{ is an odd permutation of }(1,\dots ,n)\\ 0& \text{if any two indices are equal}\end{array}$$

Key properties (independent of $n$):

• Antisymmetry under swap: interchanging any two indices flips the sign.
• Cyclic invariance (in 3D): ${ϵ}_{ijk}={ϵ}_{jki}={ϵ}_{kij}$.
• Repeated-index rule: if any two of $i_1,\dots ,i_n$ coincide, the symbol is zero.

In this module, you exclusively work in $n=3$, so the symbol is ${ϵ}_{ijk}$ - see Alternating Tensor for the full identity catalogue and worked applications.