3.2. ORIENTATION AND THE VOLUME FORM
79
or, in components,
α
j
=
E
i
1
...
i
n

1
j
v
i
1
(1)
···
v
i
n

1
(
n

1)
.
•
Theorem 3.2.8
Let
{
v
(1)
, . . . ,
v
(
n

1)
}
be an ordered
(
n

1)
tuple of lin
early independent vectors and
{
ω
(1)
, . . . , ω
(
n

1)
}
be the corresponding
1
forms. Let
α
be
1
form deﬁned by
α
=
*
±
ω
(1)
∧ ··· ∧
ω
(
n

1)
²
,
and
N
be the corresponding vector, that is,
N
i
=
g
ik
p

g

ε
i
1
...
i
n

1
k
v
i
1
(1)
···
v
i
n

1
(
n

1)
Then:
1. The vector
N
is orthogonal to all vectors
{
v
(1)
, . . . ,
v
(
n

1)
}
, that is,
(
N
,
v
(
j
)
)
=
g
ik
N
i
v
k
(
j
)
=
0
,
(
j
=
1
, . . . ,
n

1)
.
2. The ntuple
{
v
1
, . . . ,
v
n

1
,
N
}
forms a positively oriented basis.
3. The volume of the parallelepiped spanned by the vectors
{
v
(1)
, . . . ,
v
(
n

1)
,
N
}
is determined by the norm of
N
vol (
v
1
, . . . ,
v
n

1
,
N
)
=
(
N
,
N
)
.
di
ﬀ
geom.tex; April 12, 2006; 17:59; p. 81