Multivariate Calculus - Bruce E. Shapiro

LECTURE 5. VELOCITY, ACCELERATION, AND CURVATURE 41
Tangent Vectors
Our definition of the velocity is the same as the definition of the tangent vector we
in the previous section. Therefore a unit tangent vector is
ˆT =
v
‖v‖
(5.12)
Example 5.4 Find a unit tangent vector to the curve r(t) = ( 1
2 t2 , 1 3 t3 , −18t ) at
t = 1.
Solution. The velocity vector is
At t = 1,
The magnitude of the velocity is
v(t) = dr(t)
dt
= ( t, t 2 , −18 )
v(1) = (1, 1, −18)
‖v(1)‖ = √ (1) 2 + (1) 2 + (18) 2 = √ 326
Therefore
ˆT = 1 √
326
(1, 1, −18) .
Definition 5.3 The Curvature (t) is the magnitude of the rate of change of the
direction of the unit tangent vector measured with respect to distance,
κ(t) =
dT
∥ ds ∥ (5.13)
The Radius of Curvature R is the reciprocal of the curvature,
R = 1 κ
(5.14)
By the chain rule,
∥ ∥ dˆT
∥∥∥∥ κ(t) =
∥ ds ∥ = dˆT dt
∥∥∥∥ / ∥ dt ds∥ = dˆT ds
∥∥∥∥ /
dt dt ∥ = dˆT
‖v(t)‖
dt ∥
(5.15)
This gives us a more useful formula for calculating the curvature directly from the
parameterization:
κ(t) = 1
dˆT
(5.16)
‖v(t)‖ ∥ dt ∥
Math 250, Fall 2006 Revised December 6, 2006.