13.7 Cylindrical and Spherical Coordinates
Cylindrical coordinates
The cylindrical coordinates (
r, 
, z) of a point P in space are the polar coordinates (
r, ) in the xy-plane and the
usual z coordinate. They are called cylindrical because r = a is a circular cylinder about
the z-axis.
Spherical coordinates
The spherical coordinates 
of
a point P in space are defined as:
= distance from P to the origin
= same as cylindrical coordinates
= angle between the positive z-axis and
the line from P to the origin
They are called spherical coordinates because = a is a sphere about the origin with radius a.
Thus we have three different coordinate systems for three-dimensional space. The following table shows how to convert coordinates for one system to the others.
TO
FROM |
Rectangular (x, y, z) |
Cylindrical (
r, |
Spherical
|
| Rectangular (x, y, z) |
 |
 |
|
| Cylindrical (
r, |
 |
|
|
| Spherical
|
 |
 |
Problems:
1. Find the indicated coordinates of each point:
a. Spherical: 
to rectangular
b. Cylindrical: 
to spherical
c. Rectangular: 
to cylindrical
d. Spherical: 
to cylindrical
e. Cylindrical: 
to rectangular
f. Rectangular: 
to spherical
2. Identify the surface with equation:
a. 
in spherical coordinates.
b. z2 = r2 in cylindrical coordinates