DOC.
71
PRINCETON LECTURES 335
THE GENERAL THEORY
The
Riemann Tensor.
If
we
have
given
a curve
extending
from
the
point
P
to
the
point G
of
the
continuum,
then
a
vector
Au,
given
at
P, may, by
a
parallel displacement,
be
[90]
moved
along
the
curve
to G.
If
the
continuum
is
Euclidean
(more generally,
if
by
a
suitable
choice
of co-ordinates the
guv are
constants)
then the
vector
obtained
at G
as
a
result
of this
displacement
does
not depend upon
the
choice of
the
curve
joining
P and
G.
But
otherwise,
the
result
depends upon
the
path
of
the
displacement.
In
this
case,
Fig.
4.
therefore,
a
vector
suffers
a
change, AAu
(in
its direction,
not
its
magnitude),
when
it
is
carried
from
a
point
P
of
a
closed
curve,
along
the
curve,
and back
to
P.
We
shall
now
calculate
this
vector change:
=
9SÔA'~.
[91]
As
in Stokes’
theorem
for
the line
integral
of
a
vector
around
a
closed
curve,
this
problem
may
be
reduced
to
the
integration
around
a
closed
curve
with
infinitely
small
linear
dimensions;
we
shall limit
ourselves to
this
case.
[75]