Calculus: Analytic Geometry and Calculus, with Vectors

10.1 Geometry of coordinate systems 533

2 With the aid of Figure 10.12, show that the formulas giving the cylindrical
coordinates p, 0, z of a point in terms of the spherical coordinates r, 0, 0 of the
same point are
p=rsin6, .0 _ .0, z=rcos0.

3 With the aid of Problems 1 and 2, show that the formulas giving the

rectangular coordinates of a point in terms of the spherical coordinates of the

same point are

x=rcos0sin0, y=rsin0sin0, z=rcos0.

4 Transform the following equations from rectangular to polar coordinates

(a) x2 + y2 = a2 f4ns.: p = a

(b) x = a fins.: p cos ¢ = a

(c) (cos a)x + (sin a)y = a 4ns.: p cos (c& - a) = a

(d) xy = 1 fins.: p2 sin 24 = 2

(e) x2 - Y2 = a2 4ns.: p2 cos 20 = a2

5 In Problem 31 of Section 6.4, we said that the rectangular graph of the

equation

(x2 + Y2)2 = a2(x2 - Y2)

is a lemniscate. Show that the polar equation is p2 = a2 cos 2¢. The graph

appears in Figure 10.171.

6 Lemniscates have a simple geometric property. Let b be a positive num-

ber and let F, and F2 be the points (sometimes called foci) having therectangular

coordinates (-b,0) and (b,0). Let S be the set of points P for which

(1) IF,PIIF2PI = b2.

Show that the rectangular equation of S can be put in the form

(2) (x2 + Y2)2 = 2b2(x2 - y2)

and hence that S is a lemniscate. Show, in one of the various possible ways, that

the polar equation of this lemniscate is

(3) p2 = 2b2 cos 2¢.

7 Transform the following equations from polar to rectangularcoordinates.

(a) p = 3

(b) p2= a2 sin 2¢

(c) p= acos4>

(d) p = 2a(1 - cos 0)

8 Show that when

f1ns.:x2+y2=9

1ns.: (x2 + y2)2 = 2a2xy

flns:: x2 + y2 = ax

4ns.: (x2 + y2 + 2ax)2 = 4a2(x2 + Y2)

p = 2a cos 0,

the point with polar coordinates (p,4,) runs once inthe positive direction around

a curve C as ¢ increases from -a/2 toa/2. Show, in one way or another or in

more than one way, that C is the circleof radius a having its center at the point

with rectangular coordinates (a,0)