I7-10.

d~r

ds

ds

dt=

d~r

dt=~r

′ or ˆetds

dt=~r

′, |~r′|=ds

dt

d

dt

(

ˆetds

dt

)

=d

(^2) ~r
dt^2
=~r′′
ˆetd
(^2) s
dt^2
+ds
dt
dˆet
ds
ds
dt
=~r′′
ˆetd
(^2) s
dt^2

• 
  

  (ds
  dt
  )^2 κˆen=~r′′
  ~r′×~r′′=ˆet
  ds
  dt×(
  ˆetd
  (^2) s
  dt^2 + (
  ds
  dt)
  (^2) κˆen) = (ds
  dt)
  (^3) κˆeb=|~r′| (^3) κˆeb
  |~r′×~r′′|=κ|~r′|^3
  I7-11. (i)
  dφ
  ds
  =gradφ·eˆ
  (1, 1 ,1)
  =[(2xy^2 z)ˆe 1 + 2yx^2 zˆe 2 + (x^2 y^2 +x^3 )ˆe 3 ]·
  (
  3 ˆe 1 − 2 ˆe 2 + 6ˆe 3
  7
  )
  (1, 1 ,1)

  
  

  23
  7
  I7-12.
  (i) gradφ=2xyˆe 1 +x^2 ˆe 2
  dφ
  ds=gradφ·
  ˆeα= (2xyeˆ 1 +x^2 ˆe 2 )·(cosαˆe 1 + sinαˆe 2 )
  (2,√3)
  dφ
  ds
  =4
  √
  3 cosα+ 4 sinα=f(α)
  (ii) df
  dα
  =− 4
  √
  3 sinα+ 4 cosα= 0, =⇒ tanα=√^1
  3
  , =⇒ α=π/ 6 , 7 π/ 6
  f′′(α) =− 4
  √
  3 cosα−4 sinα
  f′′(π/6)< 0 maximum atπ/ 6
  f′′(7π/6)> 0 minimum at 7 π/ 6
  I7-13. Show derivative equals
  ~r·(~r′×~r′′′) +~r·(~r′′×~r′′) +~r′·(~r′×~r′′)
  where last two terms are zero. Use triple scalar product relation on last term.
  Solutions Chapter 7