Graphics Reference
In-Depth Information
The covariant derivative is a kind of directional derivative and the reader may
wonder why we do not call it just that. The reason is that this is a special case of a
concept that is defined for abstract manifolds. It is the classical terminology that was
motivated by the same reason that covariant tensors are called what they are.
The next lemma shows that the covariant derivative can be computed from any
curve g through
p
having tangent vector
v
at
p
, not just the special curve in the definition.
9.16.1. Lemma.
Let
p
Œ
R
3
and g : [a,b] Æ
R
3
be any curve with g(c) =
p
and g¢(c) =
v
. If
X
is a vector field of
R
3
defined in a neighborhood of
p
, then
()
=¢
()
—
v
Xp
X p
.
g
Proof.
Using the notation of Equation (9.91), if
()
=
(
()
()
()
)
g
q
g
q
,
g
q
,...,
g
k
q
,
1
2
then
d
dt
gt c
Dg
Ê
Ë
Ê
Ë
ˆ
ˆ
¯
¢
()
=
(
()
)
¯
()
Xp
p
,
g
g
( ( )
=—
()
∑—
()
∑
()
()
=
pp
ppv
,
,
D
g
c
(
(
—
()
∑
)
)
g
,
g
pv
,...,
g
k
pv
.
1
2
In other words,
¢
()
=
(
(
()
()
()
)
)
Xp
p
,
Dg
p
,
Dg
p
,...,
Dg
k
p
.
(9.92)
g
v
1
v
2
v
Since the right-hand side of equation (9.92) does not use the curve g but only its
tangent vector
v
at
p
, we are done.
We can use equation (9.92) to relate the covariant derivative to ordinary direc-
tional derivatives in a cleaner way, but first another definition.
Definition.
A
frame field
on a set
A
Õ
R
3
is a triple of vector fields (
U
1
,
U
2
,
U
3
) defined
over
A
with the property that, for every
p
Œ
A
,
U
1
(
p
),
U
2
(
p
), and
U
3
(
p
) are an ortho-
normal basis of the tangent space T
p
(
R
3
). The frame field (
E
1
,
E
2
,
E
3
) on
R
3
, where
E
i
(
p
) = (
p
,
e
i
), is called the
standard frame field
of
R
3
.
Let
X
be an arbitrary vector field on
R
3
and suppose that
9.16.2. Theorem.
3
Â
a
ii
i1
XE
=
=
expresses the vector field
X
in terms of the standard frame field. If
v
ΠT
p
(
R
3
), then
3
Â
()
=
() ()
—
Xp
Da
i
pE p
.
v
v
i
i
=
1
