Graphics Reference
In-Depth Information
[
]
Æ
S
h :
ab
be a curve.
Definition.
A
vector field along the curve
h is a function
X
: [a,b] Æ
R
3
with the prop-
erty that
X
(t) is a vector tangent to
S
at h(t) for all t, that is,
X
(t) Œ T
h(t)
(
S
) for all t.
The vector field
X
is said to
differentiable
if
X
(t) is a differentiable function.
Since the tangent space at a point of a surface is a vector space, it is easy to see
that the set of vector fields along a curve is a vector space. Let
X
(t) be a differentiable
vector field along h(t). Although the vector
X
¢(t) is not necessarily tangent to the
surface at h(t), its orthogonal projection onto the tangent space obviously will be.
Definition.
The
covariant derivative
of
X
(t), denoted by D
X
/dt, is the vector field
along h which sends t Œ [a,b] to the vector that is the orthogonal projection of
X
¢(t)
on the tangent space of
S
at h(t). More precisely,
D
dt
X
()
=¢
()
-¢
()
∑
(
(
()
)
)
(
( )
)
t
XXnn
t
t
h
t
h
t
,
where
n
(h(t)) is a unit normal vector to
S
at h(t).
The definition of the covariant derivative does not depend on the choice of
n
(h(t)).
9.10.15. Theorem.
Let
X
(t) and
Y
(t) be differentiable vector fields along h(t) and
let f : [a,b] Æ
R
be a differentiable function.
D
dt
D
dt
XY
D
dt
(
)
=+.
(1)
XY
+
D
dt
XX
X
f
D
dt
()
=¢ +
(2)
f
f
.
d
dt
D
dt
X
D
dt
Y
(
)
=∑
(3)
XY
∑
YX
+∑
.
Proof.
This is a straightforward computation of derivatives (Exercise 9.10.4).
Note that if
S
is a plane, then the covariant derivative is just the ordinary deriva-
tive of the vector field. Intuitively, the covariant derivative of a vector field just meas-
ures the rate of change of the vector field as seen from “inside” the surface where one
does not have any notion of a normal.
Definition.
We say that the vector field
X
(t) on h(t) is
parallel along
h if d
X
/dt =
0
,
that is, its covariant derivative vanishes.
One can easily show that if
S
is a plane, then the vector field
X
(t) is parallel along
h if and only if
X
(t) is constant. We see that we seem to have found a generalization
of what it means for vectors to be parallel. It was Levi-Civita who introduced the
covariant derivative as a means of describing parallelism. As is pointed out in