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

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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
80
CHAPTER 3. DIFFERENTIAL FORMS
3.3
Exterior Derivative
•
From now on, if not specified otherwise, we will denote the derivatives by
∂
i
=
∂
∂
x
i
.
•
The exterior derivative of a 0form (that is, a function)
f
is a 1form
d f
defined by: for any vector
v
(
d f
)(
v
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Wong
 Geometry, Derivative, Vectors, Vector Space, linearly independent vectors, Differential form, local cordinates

Click to edit the document details