Calculus: Analytic Geometry and Calculus, with Vectors

13.8 Triple integrals; spherical coordinates 711

to obtain the formula

(5) L = f b.Jr2sin20 (dl) 2 + r2(dl)2+

(dr)2

a V dt at dt dt

giving L as an integral involving spherical coordinates.

8 Using ideas and formulas from the preceding problem, start with the

formula

(1) r=xi+yj+zk

for the vector running from the origin to P at time t. Show that

(2) r = r(cos (k sin Oi + sin if sinOj + cos Ok).

Show that the velocity at time t is

dr dO d¢

(3)

where

(4) ul = cos 95 sin 0i + sin 0 sin Oj + cos Ok

(5) U2 = cos / cos 6i + sin ¢ cos Oj - sin Ok

(6) us = - sin q5i + cos q5j

Prove that the vectors ul, u2, u3, in that order, constitute a right-handed ortho-

normal system. Show, finally, that

(7) IoI = 1 v = r2 sin2 0 (d )2 l

+ r2 (a2 + \dt

)2.

(^9) With the aid of hints that may be gleaned from the two preceding prob-
lems, tell the meanings of the things in the formula
ds d (dB`/2 dr2
dt = r2 sin2 0
(dt)2

• r2 I - + Oat )
  and give conditions under which the formula is valid.
  10 This problem provides an opportunity to fill in details and gain an under-
  standing of a method by which a great problem in cosmology is solved with the
  aid of vectors and integrals that involve rectangular and spherical coordinates.
  Figure 13.891 shows a spherical ball S of radius a having its center at the origin.
  Figure 13.891