Digital Signal Processing Reference
In-Depth Information
Π
p,p
dξ
i
Γ
ij
∂
k
ξ
j
∂
j
∂
j
p
p
T
p
T
p
ξ
j
M
Fig. 17.6
Affine connections
Π
γ
(
t
)
,γ
(
t
+
dt
)
X
(
t
+
dt
)
X
(
t
)
X
(
a
)
X
(
b
)
γ
(
t
+
dt
)
γ
(
t
)
γ
(
b
)
γ
(
a
)
Fig. 17.7
Parallel translation
can be extended
to define the directional derivative along a tangent vector
D
by considering the curve
whose tangent vector is
D
. The directional derivative, denoted by
The expression (
17.3
) of the covariant derivative along a curve
γ
∇
D
X
has the
following expression:
X
p
(Γ
D
i
{
(∂
i
X
k
k
∇
D
X
=
)
p
+
ij
)
p
}
(∂
k
)
p
.
The covariant derivative along the curve
γ
can then be written as:
