Page 368 - Advanced engineering mathematics

348 CHAPTER 11 Vector Differential Calculus

Then
d d dt

G (s) = F(t(s)) = F(t)
ds dt ds
1 1

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

and this vector has a length of 1.

EXAMPLE 11.2
Let C be defined by

x = cos(t), y = sin(t), z = t/3
for −4π ≤ t ≤ 4π. C has the position vector

1
F(t) = cos(t)i + sin(t)j + tk
3
and the tangent vector
1

F (t) =−sin(t)i + cos(t)j + k.
3
√

It is routine to compute F (t) = 10/3, so the distance function along C is
1 √ 1 √
t
s(t) = 10dξ = 10(t + 4π).
−4π 3 3
In this example, we can explicitly solve for t in terms of s:

3
t = t(s) = √ s − 4π.
10
Substitute this into F(t) to get

3

G(s) = F(t(s)) = F √ s − 4π
10

3 3 1 3
= cos √ s − 4π i + sin √ s − 4π j + √ s − 4π k
10 10 3 10
3 3 1 4π

= cos √ s i + sin √ s j + √ s − k.
10 10 10 3
Now compute
3 3 3 3 1

G (s) =−√ cos √ s i + √ sin √ s j + √ k,
10 10 10 10 10
and this is a unit tangent vector to C.

Rules for differentiating various combinations of vectors are like those for functions of one
variable. If the functions and vectors are differentiable and α is a number, then

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 15, 2010 16:11 THM/NEIL Page-348 27410_11_ch11_p343-366