10
Polar, cylindrical, and spherical coordinates
10.1 Geometry of coordinate systems We begin with a glimpse of a
(or the) major reason why polar coordinates should be studied. In
many problems involving functions defined over all or portions of E3,
there is a line L which is particularly significant. This line L may, for
example, be an axis of symmetry or a wire carrying an electric current
or charge. When we want to use coordinates, we can let the line L
be the z axis of a rectangular x, y, z coordinate system as in Figure 10.11.
When we are interested in the cylinder consisting of points at a particu-
lar distance po from the z axis, we can correctly describe this set as
being the set of points having rectangular coordinates x, y, z for which
x2 + y2 = po. It is, however, much simpler to take the distance p from
the z axis to a point P to be one of the coordinates of P so that the equa-
tion of the cylinder is simply p = po. This one coordinate p is, however,
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