13Tensors and tensor fields
IA Vector Calculus
13.1 Definition
There are two ways we can think of a vector in
R
3
. We can either interpret it as
a “point” in space, or we can view it simply as a list of three numbers. However,
the list of three numbers is simply a representation of the vector with respect to
some particular basis. When we change basis, in order to represent the same
point, we will need to use a different list of three numbers. In particular, when
we perform a rotation by R
ip
, the new components of the vector is given by
v
0
i
= R
ip
v
p
.
Similarly, we can imagine a matrix as either a linear transformation or an
array of 9 numbers. Again, when we change basis, in order to represent the
same transformation, we will need a different array of numbers. This time, the
transformation is given by
A
0
ij
= R
ip
R
jq
A
pq
.
We can think about this from another angle. To define an arbitrary quantity
A
ij
, we can always just write down 9 numbers and be done with it. Moreover,
we can write down a different set of numbers in a different basis. For example,
we can define
A
ij
=
δ
ij
in our favorite basis, but
A
ij
= 0 in all other bases. We
can do so because we have the power of the pen.
However, for this
A
ij
to represent something physically meaningful, i.e. an
actual linear transformation, we have to make sure that the components of
A
ij
transform sensibly under a basis transformation. By “sensibly”, we mean that it
has to follow the transformation rule
A
0
ij
=
R
ip
R
jq
A
pq
. For example, the
A
ij
we defined in the previous paragraph does not transform sensibly. While it is
something we can define and write down, it does not correspond to anything
meaningful.
The things that transform sensibly are known as tensors. For example,
vectors and matrices (that transform according to the usual change-of-basis
rules) are tensors, but that A
ij
is not.
In general, tensors are allowed to have an arbitrary number of indices. In
order for a quantity
T
ij···k
to be a tensor, we require it to transform according to
T
0
ij···k
= R
ip
R
jq
···R
kr
T
pq···r
,
which is an obvious generalization of the rules for vectors and matrices.
Definition
(Tensor)
.
A tensor of rank
n
has components
T
ij···k
(with
n
indices)
with respect to each basis
{e
i
}
or coordinate system
{x
i
}
, and satisfies the
following rule of change of basis:
T
0
ij···k
= R
ip
R
jq
···R
kr
T
pq···r
.
Example.
–
A tensor
T
of rank 0 doesn’t transform under change of basis, and is a
scalar.
– A tensor T of rank 1 transforms under T
0
i
= R
ip
T
p
. This is a vector.
–
A tensor
T
of rank 2 transforms under
T
0
ij
=
R
ip
R
jq
T
pq
. This is a matrix.
Example.
(i) If u, v, ···w are n vectors, then
T
ij···k
= u
i
v
j
···w
k
defines a tensor of rank
n
. To check this, we check the tensor transformation
rule. We do the case for
n
= 2 for simplicity of expression, and it should
be clear that this can be trivially extended to arbitrary n:
T
0
ij
= u
0
i
v
0
j
= (R
ip
u
p
)(R
jq
v
q
)
= R
ip
R
jq
(u
p
v
q
)
= R
ip
R
jq
T
pq
Then linear combinations of such expressions are also tensors, e.g.
T
ij
=
u
i
v
j
+ a
i
b
j
for any u, v, a, b.
(ii) δ
ij
and
ε
ijk
are tensors of rank 2 and 3 respectively — with the special
property that their components are unchanged with respect to the basis
coordinate:
δ
0
ij
= R
ip
R
jq
δ
pq
= R
ip
R
jp
= δ
ij
,
since R
ip
R
jp
= (RR
T
)
ij
= I
ij
. Also
ε
0
ijk
= R
ip
R
jq
R
kr
ε
pqr
= (det R)ε
ijk
= ε
ijk
,
using results from Vectors and Matrices.
(iii)
(Physical example) In some substances, an applied electric field
E
gives rise
to a current density
j
, according to the linear relation
j
i
=
ε
ij
E
j
, where
ε
ij
is the conductivity tensor.
Note that this relation entails that the resulting current need not be in the
same direction as the electric field. This might happen if the substance
has special crystallographic directions that favours electric currents.
However, if the substance is isotropic, we have
ε
ij
=
σδ
ij
for some
σ
. In
this case, the current is parallel to the field.