A First Course in Linear Algebra, 2017a

8.2. Spherical and Cylindrical Coordinates 445
The following picture summarizes the geometric meaning of the three coordinate systems.
z
x
θ
φ
ρ
r
(ρ,φ,θ)
(r,θ,z)
(x,y,z)
(x,y,0)
y
Therefore, we can represent the same point in three ways, using Cartesian coordinates, (x,y,z), cylindrical
coordinates, (r,θ,z), and spherical coordinates (ρ,φ,θ).
Using this picture to review, call the point of interest P for convenience. The Cartesian coordinates for
P are (x,y,z). Thenρ is the distance between the origin and the point P. The angle between the positive
z axis and the line between the origin and P is denoted by φ. Then θ is the angle between the positive
x axis and the line joining the origin to the point (x,y,0) as shown. This gives the spherical coordinates,
(ρ,φ,θ). Given the line from the origin to (x,y,0), r = ρ sin(φ) is the length of this line. Thus r and
θ determine a point in the xy-plane. In other words, r and θ are the usual polar coordinates and r ≥ 0
and θ ∈ [0,2π). Letting z denote the usual z coordinate of a point in three dimensions, (r,θ,z) are the
cylindrical coordinates of P.
The relation between spherical and cylindrical coordinates is that r = ρ sin(φ) and the θ is the same
as the θ of cylindrical and polar coordinates.
We will now consider some examples.
Example 8.10: Describing a Surface in Spherical Coordinates
Express the surface z = 1 √
3
√
x 2 + y 2 in spherical coordinates.
Solution. We will use the equations from above:
x = ρ sin(φ)cos(θ),φ ∈ [0,π]
y = ρ sin(φ)sin(θ), θ ∈ [0,2π)
z = ρ cosφ, ρ ≥ 0
To express the surface in spherical coordinates, we substitute these expressions into the equation. This
is done as follows:
ρ cos(φ)= √ 1 √
(ρ sin(φ)cos(θ)) 2 +(ρ sin(φ)sin(θ)) 2 = 1 3ρ sin(φ).
3 3√
This reduces to
tan(φ)= √ 3