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[
] 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) ΠT 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
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