Calculus: Analytic Geometry and Calculus, with Vectors

532

Figure 10.171

7/4, a2 cos 20 increases from 0 to a2

and then decreases to 0. This infor-

mation enables us to sketch the two

loops of Figure 10.171, p being posi-

tive on one loop and negative on the

other. As 20 increases from 7/2 to

37/2, and hence as 0 increases from

7/4 to 37/4, cos 20 is negative and no values of p are obtained. Further

investigation shows that the graph already drawn is complete. It is

called a lemniscate.

When a > 0, the polar graph of the equation p = a0 is called a spiral

of .4rchimedes. This graph is shown in Fig. 10.181, the part for which

Figure 10.181 Figure 10.182

0 < 0 being dotted. When spirals are being graphed, and at some

other times, the approximations 7 = 3.1416, 7/2 = 1.5708, 7/4 = 0.7854,

and similar others are used. It is very often necessary to know rela-

tions akin to the relations 27 radians = 360°, jr radians = 180°, 7/2

radians = 90°, and 7/4 radians = 45°. It is sometimes useful to know

that 1 radian is 180/7 degrees or about 57 degrees, but degrees and min-

utes and seconds play minor roles in our work. The polar graph of the

equation p = eaO is an exponential spiral which is commonly called a

logarithmic spiral. The graph is shown in Figure 10.182. When a > 0,

the dotted part for which 0 < 0 spirals inward around the origin.

Problems 10.19

1 With the aid of Figure 10.11, show that the formulas giving the rectangular

coordinates x, y, z of a point in terms of the cylindrical coordinates p, 0, z of the

same point are

x=pcos0, y=psinz=z.

Polar, cylindrical, and spherical coordinates