DOC.
9
FORMAL FOUNDATION OF RELATIVITY 67
B,
when
only
linear transformations instead of
general ones
are
admitted.
[35]
Covariant
expressions
relative
to
linear
transformations.
The
algebraic properties
of
tensors
that
have
been
presented
in
§3
to
§8
are
not
simplified
if
one
limits
oneself
to
linear transformations. In
contrast,
the rules for the formation
of
tensors
by
differentiation
(§9)
become
significantly simpler.
It
applies quite generally
that
a
=
^ fas
a
a*'p
s
dx'p dxs
Consequently one
has,
for
example,
for
a
covariant
tensor
of rank
two,
according
to
(5a),
dA'ßV
_
y,
dxs
a
f
dxa
dxß
dx'p
«ßs
dx'p
dxs[dx'ßdx'v
°^l'
dx
The derivatives
--,
etc.,
are
independent
of the
xs
under linear
substitution,
ox'..
and
one
has
{15}
=
y
K
dxß
dxs
dAaß
dx'p
$
dx'ß
dx'v
dx'p
dxs
'
dA
Therefore,
|
is
a
covariant
tensor
of
rank three.
ox8
It
can
be shown in
all
generality
that
differentiating
the
components
of
any
tensor
in its coordinates
produces
a
tensor
whose rank is raised
by
one,
where the additional
index carries covariant character. In other
words,
this is the
operation
of
extension
under restriction to linear transformations. And
since
extension combined with
the
algebraic
operations
is the
very
basis of
forming
covariants,
we
have command
over
the entire
system
of covariants relative
to
linear transformations. We
now
turn
toward
[p. 1069]
deliberations that lead
to
a
much limited choice
of
coordinates.
The
transformation
law
of the
integral
J. Let H be
a
function
of
the
gßV
and
{16}
their first derivatives
-,
where
the
latter
are
called
g^v
for
short.
Now,
J
shall
be
an integral
extended
over
a
finite
part
E
of
the
continuum,
thus
J=
J
Hy/^gdr. (61)
The coordinate
system
that is used first shall be
K1.
We ask
for
the
change
AJ
of
J
when
we
go
from
system
K1
to
another
one
K2,
which
is infinitesimally
different
from
K1.
If
A6 denotes the
increase-due
to
transformation-of
an
arbitrary quantity