Calculus: Analytic Geometry and Calculus, with Vectors

10.1 Geometry of coordinate systems 527

not enough todetermine the position ofa point P. It turns out to be

convenient to determine the position of P by use of p, the angle 0, and

the rectangular coordinate z of Figure 10.11. The three numbers p,

0, z are called cylindrical coordinates of P.

z

z

-Q P'(P,O, z)

z

Y

y

Figure 10.11 Figure 10.12

In many other problems involving functions defined over all or por-
tions of E3, there is a point Po (instead of a line L) which is particularly
significant. This point Po may, for example, be the center of a spherical
or nonspherical earth or may be a point at which an electric charge is
supposed to be concentrated. When we want to use coordinates, we
can let Po be the origin 0 of a rectangular x, y, z coordinate system as in
Figure 10.12. When we are interested in the sphere consisting of points
at a particular distance ro from the origin, we can correctly describe this
set as being the set of points having rectangular coordinates x, y, z for
which x2 + y2 + z2 = ro. It is, however, much simpler to take the
distance r from the origin to a point P to be one of the coordinates of P
so that the equation of the sphere is simply r = ro. This one coordinate

r is, however, not enough to determine the position of a point P. It

turns out to be convenient to determine the position of P by use of r,
the angle 0, and the angle 0 of Figure 10.12. The three numbers r, 01
0 are called spherical coordinates of P.
We really should write P,(x,y,z), P,(p,O,z), and Pa(r,0,0) to denote,
respectively, the points having rectangular coordinates x, y, z, cylindrical
coordinates p, 0, z, and spherical coordinates r, q5, 0. The coordinates
x, y, z, p, 0, r, 0 are numbers, and it is precarious to confuse relations among
coordinates and functions of these coordinates by allowing P(0.7,0.6,0.5)
and ((0.7,0.6,0.5) to have ambiguous meanings. The subscripts are
included in Figures 10.11 and 10.12 because they should be included.
To be reasonable about this matter, we can allow P(a,b,c) to be the point
having cylindrical coordinates a, b, c while we are solving a problem or
constructing a theory in which cylindrical coordinates and no other
coordinates appear. We eliminate confusion, however, by agreeing that,