Graphics Reference
In-Depth Information
interested is the derivative of a vector field and its relation to the vector field. Let
(
U
1
,
U
2
,
U
3
) be a frame field on
R
3
. Let
v
ΠT
p
(
R
3
). Since the
U
i
define a basis at every
point,
3
Â
()
=
()
—
v
Up
c
Up
,
i
=
123
,,,
i
ij
j
j
=
1
for some c
ij
Œ
R
. It follows that
()
∑
()
c
ij
=—
v
Up Up
.
i
j
Instead of a fixed vector
v
and a varying point
p
, let us consider
p
fixed and
v
as
varying. Define functions
()
Æ£
3
()
w
ij
pRR
:
T
,
1
i j
,
£
3
,
p
by
( ()
=—
()
∑
(
.
w
ij
pv
Up Up
v
i
j
Definition.
The function w
ij
are called the
connection forms
for the frame field
(
U
1
,
U
2
,
U
3
).
The forms w
ij
have the following interpretation: If
v
=
X
(
p
), then w
ij
(
p
)(
v
) describes
the rate at which the vector field
U
i
rotates toward the vector
U
j
(
p
) as we move along
a curve through
p
with tangent vector
v
at
p
.
9.16.5. Theorem.
The connection forms are differentiable 1-forms for the manifold
R
3
and satisfy w
ij
=-w
ji
.
Proof.
To show that the w
ij
(
p
) are 1-forms one only needs to show that they are linear
on each T
p
(
R
3
) and that is easy. For the second part, note that
U
i
•
U
j
=d
ij
, so that,
using Theorem 9.16.3(4),
( ()
=
(
)( )
0 =
D
d
p UUp
Up Up Up
D
∑
v
ij
v
i
j
=
(
)( )
∑
()
+
()
∑
(
)( )
Up
v
i
j
i
v
j
( ()
+
( ()
=
w
pv
w
pv
.
ij
ij
The next theorem restates what we just showed.
9.16.6. Theorem.
Let (
U
1
,
U
2
,
U
3
) be a frame field on
R
3
and w
ij
its connection forms.
If
X
is a vector field on
R
3
, then
3
Â
()
=
() ()
(
)
( )
—
X
Up
w
pXpUp
.
(9.94)
i
ij
j
j
=
1
