Calculus: Analytic Geometry and Calculus, with Vectors

3.7 Rates, velocities 185

17 A particle P moves in E3 in such a way that the vector r running from the

origin to P is

r = r[sin 0 cos 01 + sin 0 sin q'j + cos 9k],

where r, 0, and 0 are all differentiable functions of t. Find the velocity and

speed of the particle at time t. 11ns.:

r'(t) = r'(t)u + r(t)9'(t)v + r(t)4,'(t) sin Ow

where u = sin 0 cos 4,i + sin 0 sin 4,j + cos 6k

v=cosBcos4i+cos8sin4,j - sinOk

w = - sin 4,i + cos 4,j
and
Ir'(t)I = {[r'(t)]2 + [r(t)0'(t)]2 + [r(t)q5'(t) sin 6]2}.
Remark: Problem 5 of Problems 2.69 shows that the vectors u, v, and w are
orthonormal. The numbers r, 4,, and 0 are spherical coordinates which appear
in Figure 10.12 and are studied in Chapter 10. When a is a positive number and
r = a for each t, P is always on a sphere and the above formulas become the
standard formulas used for study of curves that lie on spheres. Chapter 7 gives
solid information about curves.
18 A spherical earth has its center at the origin and has radius a. A particle
P moves on the surface S of the earth in such a way that the vector r running
from the origin to P is

(1) r = a[sin 0 cos ¢i + sin 0 sin ¢j + cos Ok]

where 0 and 4) are differentiable functions of t. Show that the velocity of P

at time t is

(2) r'(t) = aO'(t)v + a4,'(t) sin Ow

where

(3) v = cos 0 cos ci + cos 0 sin ¢j - sin Ok

(4) w = -sin ci + cos 4,j.

Remark: We invest a moment to look at some facts involving compass directions.

When 0 < 0 < 7r so that P is neither at the north pole nor at the south pole, the

vector v points south from P and the vector w points east from P. When 9 and

¢ are so related that, for some constant q,

(5) 4,'(t)sin0 = qO'(t),

the vector r'(t) and the path of P always make the same constant angle with w

and hence always have the same compass direction. The path of P is then said

to be a rhumb curve, or loxodrome. Such curves are followed by ships that keep

sailing northeast. When we have learned more calculus, we will be able to show

that (5) holds if and only if there is a constant c for which

(6) 4i=glogl-cos0+c

sin 8 or 4, = q log (csc 8 - cot B) + c

or 4,=glogtan0+c.