78
DOC.
66
MARCH
1915
First of
all,
once
again
to
the
objection
relating
to
the
special
case
that with
an
appropriate choice of coordinates
the
guv's
are
constant.[3]
You state
that
in
this
case
the
Euv’s
do not vanish
if
the
adapted
system
is selected
in
such
a
way
that the
guv's
are
not constant. But
you
did
not support
this
statement,
and
I
am
considering
it
incorrect
as
long
as
you
have not
provided
an
example
or a
general proof
for this.
Now
to
the
second
objection. Although you
concede
that
T
_(tßu A^
+
£
=
invariant,
(A^
=
ƒ
SgV'y/^dr),
where
e
is
an
infinitesimal amount of
higher
order,
you
contest,
however,
that
from
this
it
can
be concluded
that
Euv/-g
is
a
tensor.
Well,
I must
admit
that
the
basis of this
objection
is not clear
to
me.
For,
since
I
have
proven
that the
Auv's
transform
contravariantly,[4]
in
my opinion
it
is quite
irrelevant to
the
proof
that
these
Auv’s
are
infinitesimal
quantities.
Nevertheless,
an
addition
must
be made
to
the
proof
that instead
of
an
infinitesimal
tensor
Auv,
a
finite
one
is
introduced
through
the
formation of
a
limit.
I
shall not
go
into
this,
however,
but
am
convinced
that
you
will
acknowledge
the
following
derivation
as
sound:
Up
to
a
relatively
oo
small
quantity,
E

(1)
for
any justified
substitution,
as
you yourself
concede.
Up
to
a
relatively
oo
small quantity-as
you
likewise
concede-and
as
I
have
proven
in
my
last
letter,[5]
dx1 dx1Ud/T
j^l(JT
_
UJy(T
J^LV ^
dxn
dxv
(2)
From
(1)
and
(2)
it
follows,
however,
that
up
to
a
relatively
oo
small
quantity,
the
equation
V""'
&CTT
dxa
dxT
V7?
dx"
dxu
thus
also
applies.
Since
the
Auv’s
can
be chosen
independently
of
one
another,
hence
_
y dx'a
dx'T
è'aT
V-9
y
dxn
dxv
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Extracted Text (may have errors)


78
DOC.
66
MARCH
1915
First of
all,
once
again
to
the
objection
relating
to
the
special
case
that with
an
appropriate choice of coordinates
the
guv's
are
constant.[3]
You state
that
in
this
case
the
Euv’s
do not vanish
if
the
adapted
system
is selected
in
such
a
way
that the
guv's
are
not constant. But
you
did
not support
this
statement,
and
I
am
considering
it
incorrect
as
long
as
you
have not
provided
an
example
or a
general proof
for this.
Now
to
the
second
objection. Although you
concede
that
T
_(tßu A^
+
£
=
invariant,
(A^
=
ƒ
SgV'y/^dr),
where
e
is
an
infinitesimal amount of
higher
order,
you
contest,
however,
that
from
this
it
can
be concluded
that
Euv/-g
is
a
tensor.
Well,
I must
admit
that
the
basis of this
objection
is not clear
to
me.
For,
since
I
have
proven
that the
Auv's
transform
contravariantly,[4]
in
my opinion
it
is quite
irrelevant to
the
proof
that
these
Auv’s
are
infinitesimal
quantities.
Nevertheless,
an
addition
must
be made
to
the
proof
that instead
of
an
infinitesimal
tensor
Auv,
a
finite
one
is
introduced
through
the
formation of
a
limit.
I
shall not
go
into
this,
however,
but
am
convinced
that
you
will
acknowledge
the
following
derivation
as
sound:
Up
to
a
relatively
oo
small
quantity,
E

(1)
for
any justified
substitution,
as
you yourself
concede.
Up
to
a
relatively
oo
small quantity-as
you
likewise
concede-and
as
I
have
proven
in
my
last
letter,[5]
dx1 dx1Ud/T
j^l(JT
_
UJy(T
J^LV ^
dxn
dxv
(2)
From
(1)
and
(2)
it
follows,
however,
that
up
to
a
relatively
oo
small
quantity,
the
equation
V""'
&CTT
dxa
dxT
V7?
dx"
dxu
thus
also
applies.
Since
the
Auv’s
can
be chosen
independently
of
one
another,
hence
_
y dx'a
dx'T
è'aT
V-9
y
dxn
dxv

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