Calculus: Analytic Geometry and Calculus, with Vectors

10.1 Geometry of coordinate systems 529

±2, ,the numbers p, 4, + 2na constitutea set of polar coordinates

of P. We could (and sometimes do) end the matter here. It is not

always easy to know when we are being wise, but we can recognizeone

more possibility. After turning through the angle0 + a or 4, - it or

0 + (2n + 1)ir, where n is an integer, we will be facing away from P

and we can reach P by going backwards a distancep. When we take

this last possibility into account, we find that, for eachn = 0, ± 1, ±2,

. ,the numbers (-p, 0 + a + 2nlr) constitutea set of polar coordi-
nates of P. When polar coordinates of this variety are permitted to
appear in our work, we abandon the idea that p is a distance and take p
to be a coordinate that can be negative. Thus when p < 0, the point
PP(p,4) having polar coordinates p, 0 is the sameas the point Pp(I pl,
0 + tr) having the more normal polar coordinates Ipl, 4) + ir. We still

have to consider the polar coordinates of P when P is the origin 0. It

turns out to be best to agree that, for each num-

ber 4,, the numbers 0 and 4, are polar coordinates

of the origin.t

Let polar and rectangular coordinate systems be

superimposed in such a way that, as in Figure 10.14,

the initial line of the former coincides with the non-

negative x axis of the latter. When P is a point

different from 0, it is easy to obtain formulas

Figure 10.14

relating the rectangular coordinates (x,y) of P and each set (p,4)) of polar

coordinates of P for which p > 0. The definitions

• cos-x, sin

y

k--

P P

of the trigonometric functions give the formulas

(10.141) x=pcos0, y=psin0

which uniquely determine x and y in terms of p and 0. On the other

hand, the formulas

(10.142) p = x -+y2, cos q5 = x sin 0 = y

x2 + y2 x2 + y2

uniquely determine p in terms of x and y and uniquely determine an

angle 0o such that -a < 0o <it and 0 must have the form 0o + 2nr,

where n is an integer. In case p > 0 and -a/2 < 0 < a/2, the last

t In this chapter, the coordinates p, 0, r, 0 have the classical significance they usually

have in mathematical physics and elsewhere when Legendre polynomials and such things

appear. Some textbooks use r and 0 for polar coordinates and, with a shift in meaning of

coordinates, use r, 0, ¢ for spherical coordinates as we do. Sometimes p is used for a spheri-

cal coordinate. Persons who stray from one book or one classroom to another do not always

appreciate modifications of classical notation, but they are rarely if ever seriously injured.