160 DOC. 30 FOUNDATION
OF GENERAL RELATIVITY
Note.-It
is
convenient
to treat
the
scalar
(or
invariant)
both
as a
contravariant and
a
covariant
tensor of
zero
rank.
Mixed
Tensors.-We
may
also define
a
tensor
of
the
second
rank
of
the
type
Auv
=
AuBv
....
(12)
which
is covariant with
respect
to the index
u,
and contra-
variant with
respect
to
the
index
v.
Its
law
of transforma-
tion
is
bxv
bx'o"
. . .
(13)
Naturally
there
are
mixed
tensors with
any
number
of
indices of
covariant
character,
and
any
number
of indices of
contravariant character. Covariant
and contravariant
tensors
may
be looked
upon
as
special cases
of
mixed
tensors.
Symmetrical
Tensors.-A
contravariant,
or a
covariant
tensor,
of
the
second
or higher
rank
is said to be
symmetrical
if two
components,
which
are
obtained the
one
from
the other
by
the
interchange
of two
indices,
are equal.
The
tensor
Auv,
or
the
tensor
Auv,
is
thus
symmetrical
if for
any
combination
of
the
indices
u,
v,
Auv
=
Auv,
....
(14)
or respectively,
Auv
=
Avu.
....
(14a)
It has
to be
proved
that the
symmetry
thus
defined is
a
property
which is
independent
of
the
system
of reference.
It
follows
in fact
from
(9),
when
(14)
is
taken
into
consider-
ation,
that
A'or
bx'
ff
bX
T
A
(IVbX
bX
ff
bX
T
k v/*
__
bX
ff
bX
T
KflV
\'TT
=
bxT
bxy
=
bxu bxy
= bxv
bxu
=
.
The last
equation
but
one
depends upon
the
interchange
of
the summation
indices
u
and
v,
i.e. merely
on a
change
of
notation.
Antisymmetrical
Tensors.-A
contravariant
or a
covariant
tensor of
the
second,
third,
or
fourth rank
is said to
be
anti-
symmetrical
if two
components,
which
are
obtained the
one
from
the other
by
the
interchange
of two
indices,
are
equal
and
of
opposite sign.
The
tensor
Auv, or
the tensor
Auv
is
therefore
antisymmetrical,
if
always