I7-17. Vector equation of plane is(~r−~r 0 )·N~= 0where~r=xeˆ 1 +yˆe 2 +zeˆ 3 is variable
point in plane,~r 0 is fixed point in plane andN~ is normal to plane. Note thatd~r
ds=ˆetis normal to normal plane
d~r
ds×d^2 ~r
ds^2 =ˆet×κˆen=κˆebis normal to osculating plane
d^2 ~r
ds^2=κˆenis normal to rectifying planeI7-18. If~r(u,v) =x(u,v)ˆe 1 +y(u,v)ˆe 2 +z(u,v)ˆe 3 , then
∂~r
∂u=∂x
∂ueˆ 1 +∂y
∂uˆe 2 +∂z
∂uˆe 3 is tangent to coordinate curve~r(u,v 0 )
∂~r
∂v=∂x
∂vˆe 1 +∂y
∂vˆe 2 +∂z
∂veˆ 3 is tangent to coordinate curve~r(u 0 ,v)N~=∂~r
∂u×∂~r
∂vis normal to surface andˆen=
N~
|N~|=` 1 ˆe 1 +` 2 ˆe 2 +` 3 eˆ 3 is unit normal to surface where|N~|=√
(∂~r
∂u×∂~r
∂v)·(∂~r
∂u×∂~r
∂v) =√
EG−F^2I7-19. N~= gradF=
∂F
∂xˆe 1 +∂F
∂yˆe 2 +∂F
∂zˆe 3 and ˆen=|NN~~|where|N~|=HI7-20. If surface is~r=~r(x,y) =xˆe 1 +yˆe 2 +z(x,y)ˆe 3 , then
∂~r
∂x=ˆe 1 +∂z
∂xˆe 3 ∂~r
∂y=ˆe 2 +∂z
∂yˆe 3andN~ =∂~r
∂x×∂~r
∂y=−∂z
∂xˆe 1 −∂z
∂yˆe 2 +ˆe 3 with ˆen=
N~
|N~|and|N~|=√
1 + (∂x∂z)^2 + (∂z∂y)^2I7-21. ˆen=xˆe 1 +yˆe 2 or ˆen= cosθeˆ 1 + sinθˆe 2
I7-22. ˆen=xˆe 1 +yˆe 2 +zˆe 3 or ˆen= sinθcosφˆe 1 + sinθsinφˆe 2 + cosθˆe 3
I7-23. φ=ax+by+cz−d= 0andN~ = gradφ=aˆe 1 +bˆe 2 +cˆe 3 so that
ˆen=aˆe 1 +bˆe 2 +cˆe 3
√
a^2 +b^2 +c^2I7-24.^12 ˆe 1 +π 8 ˆe 2
I7-25.^1
15
ˆe 1 +^1
15ˆe 2 +^1
15ˆe 3Solutions Chapter 7