DOC. 2 COVARIANCE
PROPERTIES
13
extremum under fixed
boundary
values
of
the coordinates and their first derivatives.
It is then
possible
to
put
into the varied
manifold
M
an adapted
coordinate
system
K
which coincides with the coordinate
system
K
outside
of
L
and
deviates from K
only
inside
of
L;
because
an
extremum of
the
integral
J
must also exist for the
manifold
M
under
unchanged
boundary values-whereupon
the
satisfiability
of
the
equations
Bm
=
0 follows also for the varied manifold.
Let
us assume
the coordinate
systems
K and
K',
which
are
used in the manifold
M,
are
both
adapted. According
to
(3b)
the
equations
J'-J=
O1
+
O2,
j'
-
j
=
O1
+
O2,
or
after subtraction
(J'
-
J') -
(J
-
J)
=
(Öx
-
Ot)
+
(Ö2
-
O2).
are
valid.
The
specifications (b)
and
(c)
and the relations between M and
M,
together
with
(3), imply
that both
01
- 01
and
02
-
02
vanish.
M
can
be called
a
manifold
developed by
variation of
M.
Therefore,
we
denote
[p.
223]
analogously
J
-
J
=
8aJ,
J'
-
J'
=
SaJ',
and
consequently get
(4)
SaJ'
=
SaJ.
The index
a
is
meant to
express that,
together
with the
manifold,
the coordinate
system
is co-varied such that the varied coordinate
system
and the varied manifold
are
always adapted
relative
to
each
other,
while
on
the
boundary
the coordinate
system
remains,
of
course,
unvaried
(so-called "adapted variation").
Our aim
is
to
demonstrate that
an
equation
ÖJ = ÖJ
is
satisfied for
any
variation
of
the
manifold, not just
for
an
adapted
variation
as
equation
(4)
says.
However,
we can
let
any
variation of
guv
evolve from
an
adapted
one
if
we
follow
it
with another variation of the coordinate
system.
It
turns out
that
for
a
variation
of
guv,
equivalent
to just
one
variation
of
the coordinate
system,
the
variation
of
J,
denoted
by
SkJ,
vanishes,
provided we
assume
the variations
Sxv
and
their first derivations
vanish
on
the
boundary
of the
domain,
and also
provided
furthermore that the coordinate
system
to
be varied is
an
adapted system.
The
reason
is that
equation (3a)
leads to the direct
consequence