13Tensors and tensor fields
IA Vector Calculus
13.3 Symmetric and antisymmetric tensors
Definition
(Symmetric and anti-symmetric tensors)
.
A tensor
T
of rank
n
is
symmetric in the indices i, j if it obeys
T
ijp···q
= T
jip···q
.
It is anti-symmetric if
T
ijp···q
= −T
jip···q
.
Again, a tensor can be symmetric or anti-symmetric in any pair of indices, not
just the first two.
This is a property that holds in any coordinate systems, if it holds in one,
since
T
0
k`r...s
= R
ki
R
`j
R
rp
···R
sq
T
ijp···q
= ±R
ki
R
`j
R
rp
···R
sq
T
jip···q
= ±T
0
`kr···s
as required.
Definition
(Totally symmetric and anti-symmetric tensors)
.
A tensor is totally
(anti-)symmetric if it is (anti-)symmetric in every pair of indices.
Example. δ
ij
=
δ
ji
is totally symmetric, while
ε
ijk
=
−ε
jik
is totally antisym-
metric.
There are totally symmetric tensors of arbitrary rank n. But in R
3
,
– Any totally antisymmetric tensor of rank 3 is λε
ijk
for some scalar λ.
–
There are no totally antisymmetric tensors of rank greater than 3, except
for the trivial tensor with all components 0.
Proof: exercise (hint: pigeonhole principle)