Tensors for Physics

40 3 Symmetry of Second Rank Tensors, Cross Product

3.3 Antisymmetric Part, Vector Product

3.3.1 Dual Relation.

As already mentioned before, the antisymmetric part( 1 / 2 )(Aμν−Aνμ)of a second
rank tensorAμν, in three dimensions, can be linked with the three components of a
vectora. This link, referred to asdual relation,is:

a 1 =A 23 −A 32 = 2 Aasy 23 ,

a 2 =A 31 −A 13 = 2 Aasy 31 ,

a 3 =A 12 −A 21 = 2 Aasy 12. (3.29)

Clearly, the order of the subscript 1, 2 ,3 in the second line is a cyclic permutation
of that one in the first line, and so on.
The relation (3.29) can be inverted in the sense that, in matrix notation, the anti-
symmetric tensor is given by

Aasy=

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0

1

2

a 3 −

1

2

a 2

−

1

2

a 3 0

1

2

a 1

1

2

a 2 −

1

2

a 1 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

. (3.30)

The ‘proof’ thata, defined by (3.29) indeed transforms like a vector, whenAis a
tensor, is presented next. This is not self-evident since a vector is transformed with
one rotation matrix, whereas the second rank tensor is transformed with a product
of two transformation matrices.
The tensor property ofAimplies, that the first component ofa, in the rotated
system, is given by

a 1 ′=A 23 ′ −A′ 32 =U 2 μU 3 νAμν−U 3 μU 2 νAμν=U 2 μU 3 ν(Aμν−Aνμ). (3.31)

Note: in the last term before the last equality sign, the summation indicesμ, ν,have
been interchanged. Whena′ 1 , given by this relation is the component of a vector, the
quantitybμdefined by
bμ=Uμλ−^1 aλ′=Uλμa′λ, (3.32)

must be equal toaμ. The first component of (3.32) reads

b 1 =U 11 a′ 1 +U 21 a 2 ′+U 31 a′ 3. (3.33)