
Differential geometry 121
and
v
+Γ
2
11
u
2
+2Γ
2
12
u
v
+Γ
2
22
v
2
=0.
This result is the decoupled system of equations of the geodesic, introduced
in Section 8.1, hence the vanishing of the geodesic curvature is indeed a char-
acteristic of a geodesic curve.
Finally, since the recent discussions were mainly on parametric forms, the
equation of the geodesic for an explicitly given surface
z = z(x, y(x))
is quoted from [8] for completeness’ sake without derivation:
(1+(
∂z
∂x
)
2
+(
∂z
∂y
)
2
)
d
2
y
dx
2
=
∂z
∂x
∂
2
z
∂y
2
(
dy
dx
)
3
+
(2
∂z
∂x
∂
2
z
∂x∂y
−
∂z
∂y
∂
2
z
∂y
2
)(
dy
dx
)
2
+
(
∂z
∂x
∂
2
z
∂x
2
− 2
∂z
∂y
∂
2
z
∂x∂y
)
dy
dx
−
∂z
∂y
∂
2
z
∂x
2
.
The formula is rather overwhelming andusefulonlyinconnectionwiththe
simplest surfaces. ...