Calculus: Analytic Geometry and Calculus, with Vectors

528 Polar, cylindrical, and spherical coordinates

except in special situations where there is an explicit agreement to the

contrary, P(a,b,c) is always the point having rectangular coordinates

a, b, c.

In ordinary useful applications of the cylindrical coordinates of Figure

10.11, it is always supposed that p > 0 and it is sometimes supposed

that p > 0 and -7r < , < a. Sometimes the latter restriction on 0 is

removed so that q can vary continuously as a particle having cylindrical

coordinates (p,4,,z) makes excursions around the z axis. In those situa-

tions in which it is supposed that p > 0 or p >_ 0, the graph in cylindrical

coordinates of the equation 0 = 4o is a half-plane (not a whole plane)

having an edge on the z axis. In those situations in which 0 is unre-

stricted, a point P does not determine its cylindrical coordinates because

the two points

P, (p, 0 + 2na, z), P,(p, P, z)

are identical when n is an integer. In ordinary useful applications of

spherical coordinates, it is always supposed that r > 0 and is sometimes

supposed that r > 0, 0 5 0<--_ir, and --7r < 0 - ir. In ordinary geo-

graphical terms the coordinate 0, which is 0 when P,(r,0,0) is at the north

pole and is 7r/2 when P,(r,qS,8) is on the equator and is a when P,(r,0,0)

is at the south pole, determines the latitude of P,(r,0,0). The coordinate

cp determines the longitude.

Partly because the endeavor helps us to understand cylindrical and

spherical coordinates, we turn to the study of polar coordinates of points

in a plane. The basic idea behind the concept of polar coordinates is

both simple and attractive. Suppose we are located at a point 0, an

origin or pole, in a plane and we wish to give explicit instructions telling

how to make a pilgrimage to a point P in the same plane. We begin by

Figure 10.13

constructing a half-line OA with an end at 0 as in

Figure 10.13 and calling this half-line the initial line

from which angles are to be measured. In case P is

not the origin, instructions for reaching P are now

easily given. Start at the origin looking in the direc-

tion of the initial line, turn in the positive (counter-

clockwise) direction through the angle 0 until facing P,

and then travel the appropriate distance p from 0 to P. We could (and

sometimes do) end the matter here and say that p and 0 are the polar

coordinates of P,(p,¢), the point having polar coordinates p and 0.

Sometimes we restrict 0 to the domain -,r < .0 <9r and end the matter

in another way. While recognition of the fact is sometimes irksome, it is

nevertheless true that if n is an integer which may be negativeas well

as positive or zero and if we turn through the angle 2n7r + 0, then we

will be facing toward P and can travel the distancep to reach P. When

we take this possibility into account, we find that, for each n = 0, ± 1,