7-17. Let r (s)denote the position vector of a space curve which is defined in terms
of the arc length s.
(a) Show that the equation of the rectifying plane can be written as
(r (s)−r (s 0 )) ·d
(^2) r (s 0 )
dx^2
= 0
(b) Show that the equation of the osculating plane can be written as
[r (s)−r (s 0 )]·
[
dr(s 0 )
ds
×d
(^2) r (s 0 )
ds^2
]
= 0
(c) Show that the equation of the normal plane can be written as
[r (s)−r (s 0 )] ·
dr(s 0 )
ds = 0
7-18. Show that the direction cosines ( 1 , 2 , 3 )of the normal to the surface
r =r (u, v )are given by
1 =
∣∣
∣∣
∂y
∂u
∂z
∂y ∂u
∂v
∂z
∂v
∣∣
∣∣
D
, 2 =
∣∣
∣∣
∂z
∂u
∂x
∂z ∂u
∂v
∂x
∂v
∣∣
∣∣
D
, 3 =
∣∣
∣∣
∂x
∂u
∂y
∂x ∂u
∂v
∂y
∂v
∣∣
∣∣
D
,
where
D=
√
EG −F^2.
7-19. Show that the direction cosines ( 1 , 2 , 3 ) of the normal to the surface
F(x, y, z) = 0 are given by
1 =
∂F
∂x
H, ^2 =
∂F
∂y
H, ^3 =
∂F
∂z
H,
where
H^2 =
(
∂F
∂x
) 2
+
(
∂F
∂y
) 2
+
(
∂F
∂z
) 2
.
7-20. Show that the direction cosines ( 1 , 2 , 3 )of the normal to the surface
z=z(x, y )are given by
1 =
−∂z∂x
H , ^2 =
−∂z∂y
H , ^3 =
1
H,
where
H^2 =
(
∂z
∂x
) 2
+
(
∂z
∂y
) 2
+ 1