13Tensors and tensor fields
IA Vector Calculus
13.4 Tensors, multi-linear maps and the quotient rule
Tensors as multi-linear maps
In Vectors and Matrices, we know that matrices are linear maps. We will prove
an analogous fact for tensors.
Definition
(Multilinear map)
.
A map
T
that maps
n
vectors
a, b, ··· , c
to
R
is multi-linear if it is linear in each of the vectors a, b, ··· , c individually.
We will show that a tensor
T
of rank
n
is a equivalent to a multi-linear map
from n vectors a, b, ··· , c to R defined by
T (a, b, ··· , c) = T
ij···k
a
i
b
j
···c
k
.
To show that tensors are equivalent to multi-linear maps, we have to show the
following:
(i)
Defining a map with a tensor makes sense, i.e. the expression
T
ij···k
a
i
b
j
···c
k
is the same regardless of the basis chosen;
(ii)
While it is always possible to write a multi-linear map as
T
ij···k
a
i
b
j
···c
k
,
we have to show that
T
ij···k
is indeed a tensor, i.e. transform according to
the tensor transformation rules.
To show the first property, just note that the
T
ij···k
a
i
b
j
···c
k
is a tensor
product (followed by contraction), which retains tensor-ness. So it is also a
tensor. In particular, it is a rank 0 tensor, i.e. a scalar, which is independent of
the basis.
To show the second property, assuming that
T
is a multi-linear map, it must
be independent of the basis, so
T
ij···k
a
i
b
j
···c
k
= T
0
ij···k
a
0
i
b
0
j
···c
0
k
.
Since
v
0
p
=
R
pi
v
i
by tensor transformation rules, multiplying both sides by
R
pi
gives v
i
= R
pi
v
0
p
. Substituting in gives
T
ij···k
(R
pi
a
0
p
)(R
qj
b
0
q
) ···(R
kr
c
0
r
) = T
0
pq···r
a
0
p
b
0
q
···c
0
r
.
Since this is true for all a, b, ···c, we must have
T
ij···k
R
pi
R
qj
···R
rk
= T
0
pq···r
Hence T
ij···k
obeys the tensor transformation rule, and is a tensor.
This shows that there is a one-to-one correspondence between tensors of rank
n and multi-linear maps.
This gives a way of thinking about tensors independent of any coordinate
system or choice of basis, and the tensor transformation rule emerges naturally.
Note that the above is exactly what we did with linear maps and matrices.
The quotient rule
If T
i ···j
|{z}
n
p ···q
|{z}
m
is a tensor of rank n + m, and u
p···q
is a tensor of rank m then
v
i,···j
= T
i···jp···q
u
p···q
is a tensor of rank
n
, since it is a tensor product of
T
and
u
, followed by
contraction.
The converse is also true:
Proposition
(Quotient rule)
.
Suppose that
T
i···jp···q
is an array defined in each
coordinate system, and that
v
i···j
=
T
i···jp···q
u
p···q
is also a tensor for any tensor
u
p···q
. Then T
i···jp···q
is also a tensor.
Note that we have previously seen the special case of
n
=
m
= 1, which says
that linear maps are tensors.
Proof.
We can check the tensor transformation rule directly. However, we can
reuse the result above to save some writing.
Consider the special form
u
p···q
=
c
p
···d
q
for any vectors
c, ···d
. By
assumption,
v
i···j
= T
i···jp···q
c
p
···d
q
is a tensor. Then
v
i···j
a
i
···b
j
= T
i···jp···q
a
i
···b
j
c
p
···d
q
is a scalar for any vectors
a, ··· , b, c, ··· , d
. Since
T
i···jp···q
a
i
···b
j
c
p
···d
q
is a
scalar and hence gives the same result in every coordinate system,
T
i···jp···q
is a
multi-linear map. So T
i···jp···q
is a tensor.