13Tensors and tensor fields
IA Vector Calculus
13.2 Tensor algebra
Definition
(Tensor addition)
.
Tensors
T
and
S
of the same rank can be added;
T + S is also a tensor of the same rank, defined as
(T + S)
ij···k
= T
ij···k
+ S
ij···k
.
in any coordinate system.
To check that this is a tensor, we check the transformation rule. Again, we
only show for n = 2:
(T + S)
0
ij
= T
0
ij
+ S
0
ij
= R
ip
R
jq
T
pq
+ R
ip
R
jq
S
pq
= (R
ip
R
jq
)(T
pq
+ S
pq
).
Definition
(Scalar multiplication)
.
A tensor
T
of rank
n
can be multiplied by
a scalar α. αT is a tensor of the same rank, defined by
(αT )
ij
= αT
ij
.
It is trivial to check that the resulting object is indeed a tensor.
Definition
(Tensor product)
.
Let
T
be a tensor of rank
n
and
S
be a tensor of
rank m. The tensor product T ⊗ S is a tensor of rank n + m defined by
(T ⊗ S)
x
1
x
2
···x
n
y
1
y
2
···y
m
= T
x
1
x
2
···x
n
S
y
1
y
2
···y
n
.
It is trivial to show that this is a tensor.
We can similarly define tensor products for any (positive integer) number of
tensors, e.g. for n vectors u, v ··· , w, we can define
T = u ⊗ v ⊗ ··· ⊗ w
by
T
ij···k
= u
i
v
j
···w
k
,
as defined in the example in the beginning of the chapter.
Definition
(Tensor contraction)
.
For a tensor
T
of rank
n
with components
T
ijp···q
, we can contract on the indices
i, j
to obtain a new tensor of rank
n −
2:
S
p···q
= δ
ij
T
ijp···q
= T
iip···q
Note that we don’t have to always contract on the first two indices. We can
contract any pair we like.
To check that contraction produces a tensor, we take the ranks 2
T
ij
example.
Contracting, we get
T
ii
,a rank-0 scalar. We have
T
0
ii
=
R
ip
R
iq
T
pq
=
δ
pq
T
pq
=
T
pp
= T
ii
, since R is an orthogonal matrix.
If we view
T
ij
as a matrix, then the contraction is simply the trace of
the matrix. So our result above says that the trace is invariant under basis
transformations — as we already know in IA Vectors and Matrices.
Note that our usual matrix product can be formed by first applying a tensor
product to obtain M
ij
N
pq
, then contract with δ
jp
to obtain M
ij
N
jq
.