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 = 07-18. 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