Michael Corral: Vector Calculus

64 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE
Exercises 7-9 develop the moving frame field T, N, B at a point on a curve.
7. Let f(t) be a smooth curve such that f ′ (t)0for all t. Then we can define the unit
tangent vector T by
T(t)= f′ (t)
‖f ′ (t)‖ .
Show that
T ′ (t)= f′ (t)×(f ′′ (t)×f ′ (t))
‖f ′ (t)‖ 3 .
8. Continuing Exercise 7, assume that f ′ (t) and f ′′ (t) are not parallel. Then T ′ (t)0
so we can define the unit principal normal vector N by
Show that
N(t)= T′ (t)
‖T ′ (t)‖ .
N(t)= f′ (t)×(f ′′ (t)×f ′ (t))
‖f ′ (t)‖‖f ′′ (t)×f ′ (t)‖ .
9. Continuing Exercise 8, the unit binormal vector B is defined by
Show that
B(t)=T(t)×N(t).
B(t)= f′ (t)×f ′′ (t)
‖f ′ (t)×f ′′ (t)‖ .
Note: The vectors T(t), N(t) and B(t) form a right-handed system of mutually perpendicular
unit vectors (called orthonormal vectors) at each point on the curve f(t).
10. Continuing Exercise 9, the curvatureκis defined by
Show that
κ(t)= ‖T′ (t)‖
‖f ′ (t)‖ =‖f′ (t)×(f ′′ (t)×f ′ (t))‖
‖f ′ (t)‖ 4 .
κ(t)= ‖f′ (t)×f ′′ (t)‖
‖f ′ (t)‖ 3
and that T ′ (t)=‖f ′ (t)‖κ(t)N(t).
Note:κ(t) gives a sense of how “curved” the curve f(t) is at each point.
11. Find T, N, B andκat each point of the helix f(t)=(cost,sint,t).
12. Show that the arc length L of a curve whose spherical coordinates areρ=ρ(t),
θ=θ(t) andφ=φ(t) for t in an interval [a,b] is
∫ b √
L= ρ ′ (t) 2 +(ρ(t) 2 sin 2 φ(t))θ ′ (t) 2 +ρ(t) 2 φ ′ (t) 2 dt.
a