DOC.
71
PRINCETON LECTURES 329
THE GENERAL THEORY
are, unfortunately, considerably
more
complicated.
The
reason
for this
is
as
follows.
If
Au
is
a
contra-variant
vector,
the
coefficients
of
its
transformation,
dx'u/dxv,
are
in-
dependent
of
position
only
if
the transformation
is
a
linear
one.
Then the
vector components,
Au
+
dAu/dxadxa,
at
a
neighbouring point
transform in the
same
way
as
the
Au,
from
which
follows
the
vector
character of the
vector
differentials,
and the
tensor
character
of
dAu/dxa.
But if
the
dx'u/dxv
are
variable
this
is
no
longer true.
That there
are, nevertheless,
in
the
general case,
invari-
ant
differential
operations
for
tensors,
is recognized most
satisfactorily
in
the
following
way,
introduced
by
Levi-
Civita
and
Weyl.
Let
(Au)
be
a
contra-variant
vector
whose
components
are given
with
respect to
the
co-ordinate
system
of the
xv.
Let
P1
and
P2
be
two infinitesimally
near
points
of the continuum. For the infinitesimal
region
surrounding
the
point
P1,
there
is,
according to
our
way
of
considering
the matter,
a
co-ordinate
system
of
the
Xv
(with
imaginary
X4-co-ordinate
)
for
which the continuum
is
Euclidean. Let
Au(1)
be
the co-ordinates of
the
vector
at
the
point
P1.
Imagine
a
vector
drawn
at
the
point
P2,
using
the
local
system
of
the
Xv,
with the
same
co-ordinates
(parallel
vector through
P2),
then
this
parallel vector
is
uniquely
determined
by
the
vector at
P1
and the
displace-
ment.
We
designate
this
operation,
whose
uniqueness
will
appear
in the
sequel,
the
parallel displacement
of
the
vector
Au
from
P1
to
the
infinitesimally near
point
P2.
If
we
form
the
vector
difference
of
the
vector
(Au)
at
the
point
P2
and the
vector
obtained
by parallel displacement
from
[86]
[69]