172
DOC. 30 FOUNDATION OF GENERAL RELATIVITY
type
AuBv.
It
will
therefore
be sufficient
to
deduce
the
ex-
pression
for
the
extension
of
a
tensor
of
this
special type.
By
(26)
the
expressions
^
t}A"
r}Br,
are
tensors.
On
outer
multiplication
of
the
first
by Bv,
and
of
the
second
by
Au,
we
obtain
in each
case a
tensor of
the
third
rank.
By adding these,
we
have the tensor
of
the third
rank
7)\
A^r
=
-
{cr/x,
T}At"
- {TV,
TJA^T
. .
(27)
where
we
have
put
Auv =
AuBv.
As the
right-hand
side
of
(27)
is linear
and
homogeneous
in the
Auv
and
their first
derivatives,
this
law of
formation
leads to
a
tensor,
not
only
in
the
case
of
a
tensor of
the
type
AuBv,
but
also
in the
case
of
a sum
of such
tensors,
i.e.
in the
case
of
any
covariant
tensor of
the
second
rank.
We call
Auvo
the extension
of
the
tensor
Auv.
It
is
clear that
(26)
and
(24) concern only special
cases
of
extension
(the
extension
of
the
tensors of
rank
one
and
zero
respectively).
In
general,
all
special
laws
of
formation
of
tensors
are
in-
cluded
in
(27)
in
combination with the
multiplication
of
tensors.
§
11.
Some Cases of
Special
Importance
The
Fundamental
Tensor.-We will first
prove' some
lemmas
which
will be useful
hereafter.
By
the rule
for
the
differentiation
of
determinants
dg
=
guvgdguv
= -
guvgdguv
.
.
(28)
The last
member is
obtained
from
the last
but
one,
if
we
bear
in mind that
guvgu'v
=
Su'u,
so
that
guvguv
=
4,
and
conse-
quently
guvdguv
+
guvdguv
=
0.