Page 370 - Advanced engineering mathematics

350 CHAPTER 11 Vector Differential Calculus

Then
ds

v(t) = F (t) = ,
dt
which is the rate of change with respect to time of the distance along the trajectory or path of
motion.

The acceleration a(t) is the rate of change of the velocity with respect to time, or
a(t) = v (t) = F (t).

If F (t) = O, then this vector is a tangent vector to C. We obtain a unit tangent vector T(t)

by dividing F (t) by its length. This leads to various expressions for the unit tangent vector to C:

1 1

T(t) = F (t) = F (t)
F (t) ds/dt

1 1
= v(t) = v(t).
v(t) v(t)
Thus, the unit tangent vector is also the velocity vector divided by the speed.
The curvature κ(s) of C is defined as the magnitude of the rate of change of the unit tangent
with respect to arc length along C:

.
dT
ds
κ(s) =
This definition is motivated by Figure 11.4, which suggests that the more a curve bends at
a point, the faster the unit tangent vector is changing direction there. This expression for the
curvature, however, is difficult to work with because we usually have the unit tangent vector as a

z

y

x

FIGURE 11.4 Curvature as a rate of
change of the tangent vector.

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October 15, 2010 16:11 THM/NEIL Page-350 27410_11_ch11_p343-366