The volume form.
1. Orientation on vector spaces and manifolds.
(1) Let
V
be a vector space of dimension
n
≥
1. Recall that an
orientation
on
V
is speciﬁed by choosing a particular basis
e
1
, . . . , e
n
of
V
. If
v
1
, . . . , v
n
is any other basis of
V
, this basis is declared
positive
if the transition matrix
A
has
det
(
A
)
>
0 and
negative
if
det
(
A
)
<
0.
Here
A
= (
v
j
i
)
ij
, where
v
i
=
∑
n
j
=1
v
j
i
e
j
. Note that if
v
1
=
e
1
, . . . v
n
=
e
n
,
then
A
=
I
n
is the identity matrix and hence
e
1
, . . . , e
n
is a positive basis.
(2) Let
M
n
be an
n
manifold. An
orientation
of
M
can be viewed as
a continuous (with respect to varying
p
) choice of orientations on all the
tangent spaces
M
p
, where
p
∈
M
.
Let (
U
i
, φ
i
)
i
be an orienting atlas on
M
(that is the Jacobians of all the
transition maps
φ
i
◦
φ

1
j
have positive determinants). For
p
∈
M
this atlas
deﬁnes an orientation on
M
p
by declaring that the basis
∂
∂x
1
i

p
, . . . ,
∂
∂x
n
i

p
is
a positive basis of
M
p
, where
p
∈
(
U
i
, φ
i
= (
x
1
i
, . . . , x
n
i
)).
Conversely, suppose that we have speciﬁed a a continuous (with respect to
varying
p
) choice of orientations on all the tangent spaces
M
p
, where
p
∈
M
.
We can construct an orienting atlas on
M
as follows. Start with any atlas
(
U
i
, φ
i
= (
x
1
i
, . . . , x
n
i
))
i
with connected sets
U
i
. For every
U
i
choose
p
∈
U
i
and check whether
∂
∂x
1
i

p
, . . . ,
∂
∂x
n
i

p
is a positive basis of
M
p
. If yes, keep
(
U
i
, φ
i
) without change. If not, interchange
x
1
i
and
x
2
i
in
φ
i
. Do this for
every
i
. The resulting collection of charts is an orienting atlas on
M
.
2. The volume form on a vector space.
Let
V
be a vector space of
dimension
n
≥
1 with a chosen orientation. Let
h
,
i
be a positivedeﬁnite
inner product on
V
. The
volume form
on
V
corresponding to this choice of