174
DOC. 30 FOUNDATION OF GENERAL RELATIVITY
In
accordance
with
(31)
and
(29),
the
last term
of
this
ex-
pression may
be
written
&
+
Ä,
+
1
-
gg^k.
bx,
bx
J
-
9
bx
As
the
symbols
of
the
indices of summation
are
immaterial,
the
first
two terms
of
this
expression
cancel
the
second of
the
one
above.
If
we
then write
guvAu
=
Av,
so
that
Av
like
Au
is
an arbitrary vector, we
finally
obtain
1
lW~^gk').
. .
(35)
0
=
J~^~g
lx,
This
scalar
is
the
divergence
of
the contravariant vector
Av.
The
"Curl"
of
a
Covariant Vector.-The
second
term in
(26)
is
symmetrical
in the
indices
u
and
v.
Therefore
Auv
-
Avu
is
a
particularly
simply
constructed
antisym-
metrical tensor.
We obtain
B
=
^
bxv
bK
bx.
(36)
Antisymmetrical
Extension
of
a
Six-vector.-Applying
(27)
to
an
antisymmetrical
tensor
of
the
second
rank
Auv,
forming
in
addition the two
equations
which arise
through
cyclic
permutations
of
the
indices,
and
adding
these
three
equations,
we
obtain the tensor
of
the third rank
Aiii/er
"t"
Awru
A(ffiv
ruv - •fiver FTfl
bApy
^
bAvtr
bx bx
^
bA-rfx
bX
(37)
which it
is
easy
to
prove
is
antisymmetrical.
The
Divergence of a
Six-vector.-Taking
the
mixed
pro-
duct
of
(27) by
guagvß,
we
also
obtain
a
tensor.
The
first
term
on
the
right-hand
side of
(27)
may
be
written in the
form
0
(g^g'fik^)
-
gt^Kv
-
g-f&^kjlV*bx,
bx
bx,
If
we
write
AaoB
for
guagvBAuvo
and
AaB
for
guagvBAuv,
and
in
the
transformed first term
replace
V*
and
»9"
bX
bx
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