4.1 Definition, Properties 49
εμνλεμ′ν′λ′=∣ ∣ ∣ ∣ ∣ ∣
δμμ′δμν′δμλ′
δνμ′δνν′δνλ′
δλμ′δλν′δλλ′∣ ∣ ∣ ∣ ∣ ∣
. (4.8)
The rows and columns in the determinant can be interchanged and one has
εμνλεμ′ν′λ′=εμ′ν′λ′εμνλ. Written explicitly, (4.8) is equivalent to
εμνλεμ′ν′λ′=δμμ′δνν′δλλ′+δμν′δνλ′δλμ′+δμλ′δνμ′δλν′
−δμμ′δνλ′δλν′−δμν′δνμ′δλλ′−δμλ′δνν′δλμ′. (4.9)The relation (4.8) can be inferred from (4.1) as follows. First, notice that the deter-
minant of the product of two matrices is equal to the product of the determinants
of these matrices. Furthermore, in the defining relation (4.1)forεμ′ν′λ′,therows
and columns are interchanged in the determinant. Thenεμ′ν′λ′εμνλis equal to the
determinant of the matrix product
⎛
⎝δ 1 μ′δ 2 μ′δ 3 μ′
δ 1 ν′δ 2 ν′δ 3 ν′
δ 1 λ′δ 2 λ′δ 3 λ′⎞
⎠
⎛
⎝
δ 1 μδ 1 νδ 1 λ
δ 2 μδ 2 νδ 2 λ
δ 3 μδ 3 νδ 3 λ⎞
⎠.
Matrix multiplication, row times column, yieldsδ 1 μ′δ 1 μ+δ 2 μ′δ 2 μ+δ 3 μ′δ 3 μ =
δκμ′δκμ = δμ′μ = δμμ′for the 11-element. Similarly, the other elements are
obtained, as they appear in the determinant (4.8).
In many applications, a contracted version of the product of two epsilon-tensors
is needed where two subscripts are equal and summed over. In particular, forλ=λ′,
(4.8) and (4.9) reduce to
εμνλεμ′ν′λ=∣
∣
∣
∣
δμμ′δμν′
δνμ′δνν′∣
∣
∣
∣=δμμ′δνν′−δμν′δνμ′. (4.10)Notice thatεμνλεμ′ν′λ=εμνλελμ′ν′. Formulae known for the product of two vector
products follow from (4.10).
The further contraction of (4.10), withν=ν′, yields
εμνλεμ′νλ= 2 δμμ′. (4.11)The total contraction of two epsilon-tensors is equal to 6, viz.:
εμνλεμνλ= 6. (4.12)This value is equal to the number of non-zero elements of the epsilon-tensor in 3D.
Side remark: by analogy, a totally antisymmetric isotropic tensor can be defined in
D dimensions. Then the total contraction of this tensor with itself, corresponding to
(4.12) is D!. Clearly, this value is 6 for D=3.