48 4 Epsilon-Tensor
The epsilon-tensor is totally antisymmetric, i.e. it changes sign, when two indices
are interchanged. It is equal to zero, when two indices are equal. Furthermore, the
tensorεμνλis isotropic. This means, just like the unit tensorδμν, it is form-invariant
upon a rotation of the coordinate system.
The dual relation between a vector and the antisymmetric part of a tensor, as given
by (3.29), is equivalent to
aμ=εμνλAνλ. (4.3)
To verify this relation, consider the 1-component. Then one hasa 1 =ε 1 νλAνλ=
ε 123 A 23 +ε 132 A 32 =A 23 −A 32. Notice that the positions of the summation indices
matter, not their names. The double contraction of the epsilon-tensor with a second
rank tensor, as in (4.3), projects out the antisymmetric part of the tensor, i.e.
εμνλAνλ=εμνλAasyνλ. (4.4)This can be seen as follows. Trivially, since 1=^12 +^12 , one hasεμνλAνλ =
1
2 εμνλAνλ+
1
2 εμνλAνλ. The renamingν, λ→ λνof the summation indices in
the second term on the right hand side, leads toεμνλAνλ=^12 εμνλAνλ+^12 εμλνAλν.
Next,εμλν =−εμνλis used. ThenεμνλAνλ=^12 εμνλ(Aνλ−Aλν)is obtained,
which corresponds to (4.4).
By analogy to (4.3), the vector product of two vectorsaandb, defined by (3.38),
can be written as
cμ=εμνλaνbλ. (4.5)
The properties of the vector product discussed above follow from the properties of
the epsilon-tensor.
The spate productd·(a×b)corresponds toεμνλdμaνbλ. A cyclic renaming of
the summation indices and the use ofεμνλ =ενλμ=ελμνimplies
εμνλaμbνdλ=εμνλbμdνaλ, (4.6)or, equivalently
d·(a×b)=a·(b×d)=b·(d×a). (4.7)
Of course, the symmetry of the spate product can also be inferred from the symmetry
properties of the determinant shown in (3.41).
4.1.2 Product of Two Epsilon-Tensors.
The product of two epsilon-tensors is a tensor of rank 6 which can be expressed in
terms of triple products of the unit second rank tensor, in particular