Tensors for Physics

11.3 Generalized Cross Product,-Tensors 187

Letwbe a vector andAa tensor of rank. The generalized cross product is
given by

(w×A)μ 1 μ 2 ···μ=()μ 1 μ 2 ···μ,λ,μ′

1 μ

′

2 ···μ

′



wλAμ′ 1 μ′ 2 ···μ′. (11.17)

For a symmetric traceless tensorS, the generalized cross product can also be writ-
ten as

(w×S)μ 1 μ 2 ···μ=()μ
1 μ 2 ···μ,λ,μ′ 1 μ′ 2 ···μ′
wλSμ′ 1 μ′ 2 ···μ′=εμ 1 λνwλSνμ 2 ···μ.
(11.18)
For=1, one has

(μ,λ,ν^1 ) =εμλν.

Thus, in this case, (11.17) and (11.18) correspond to the standard vector product,
cf. Sects.3.3and 4 .For=2, the-tensor is explicitly expressed by a linear
combination of products of epsilon- and delta-tensors, viz.

(μν,λ,μ^2 ) ′ν′≡μν,λ,μ′ν′=

1

4

(εμλμ′δνν′+εμλν′δνμ′+ενλμ′δμν′+ενλν′δμμ′).

(11.19)

Thepropertiesμμ,λ,μ′ν′=0andμν,λ,μ′μ′=0followfromtheexplicitexpression
given above, due to the antisymmetry of the epsilon-tensor, e.g.εν′λμ′=−εμ′λν′.
The contractionμ′=μ,in(11.19), yields

μν,λ,μν′=

5

4

ενλν′. (11.20)

The cross product of a vectorwwith a symmetric traceless second rank tensorSis,
in accord with (11.18), given by

(w×S)μν=μν,λ,μ′ν′wλSμ′ν′=εμλκwλSκν=

1

2

(εμλκwλSκν+ενλκwλSκμ).

(11.21)

As an alternative to (11.16), the()-tensor can also be defined in terms of the
epsilon-tensor and oneΔ-tensor of rank+1, more specifically:

()μ 1 μ 2 ···μ,λ,μ′

1 μ′ 2 ···μ′

=

+ 1



2 + 1

2 + 3

Δμ( 1 +μ^12 )···μμ,μ′μ′

1 μ′ 2 ···μ′

εμμ′λ. (11.22)