180
DOC.
13
GENERALIZED THEORY OF RELATIVITY
(33)
these
two
differential
operations
coincide
in
the
case
of
symmetric
tensors.
Since
(34)
formula
(33) can
also
be rewritten
as
(35)
§3.
Special
Tensors
(Vectors)
We shall call
a
covariant
(contravariant)
tensor
special
if
its
components
form
a
system
of
alternating functions
of the basic variables.
Accordingly,
the
components
of
a
special
tensor
are
subject
to
the
following
conditions:
1.
Tr
r2 r
=
0
if
two
of the indices
r1,
r2,...rx
are
identical.
1
A
2.
If
r1,
r2,...rx
and
s1,
s2,...sx
differ
only
in the
sequence
of
the indices,
then
,
depending
on
whether
r1,
r2,...rx
and
s1,
s2,...sx are
permutations
of the
same
class
or
not.
As
we
know, two
permutations belong
to
the
same
class if both
are
formed from the basic
permutation
1, 2,
...
n
by
means
of
an
even or an
odd number of
mere
interchanges
of
two
indices.
The number
of
linearly independent components
of
a special
tensor
of
rank
A
is
T
=
±T
r\r2-"r\
s\s2---s\
thus
n
A
\
/
Owing
to
these
properties,
the
theory
of
special
tensors turns out to
be
simpler
but also richer than that of
general
tensors; it is
of
special significance
for mathemati-
cal
physics
because the
theory
of
vectors of
the
X-th
kind
(four-vectors,
six-vectors
for
n
=
4) can
be reduced to
the
special
tensors
of
rank
A.
From
the
standpoint
of
the
general theory,
it is
more
expedient
to start out
from
tensors
and
to treat vectors
merely as special
tensors.
In the
vector
analysis
of the n-dimensional manifold
k2
=
E
**
/XV
an
important
role
is
played
by a
special
tensor
of the nth rank that is connected with
(as,.5
1rk~
frkl
=
E J
`-S
irs
rk
S
`.3
1
ag,.3
2~'~
aXk
=
aiogy~
aXk
nc
+
=
1
E
r
rk
aXk
=
aiogy~
aXk
nc
+
= 1
E
r
rk