Calculus- Early Transcendentals, 2021a

448 Three Dimensions
Example 12.16
Find an equation for the sphere of radius 2 in spherical coordinates.
Solution. If we start with the Cartesian equation of the sphere and substitute, we get the spherical equation:
x 2 + y 2 + z 2 = 2 2
ρ 2 sin 2 φ cos 2 θ + ρ 2 sin 2 φ sin 2 θ + ρ 2 cos 2 φ = 2 2
ρ 2 sin 2 φ(cos 2 θ + sin 2 θ)+ρ 2 cos 2 φ = 2 2
ρ 2 sin 2 φ + ρ 2 cos 2 φ = 2 2
ρ 2 (sin 2 φ + cos 2 φ)=2 2
ρ 2 = 2 2
ρ = 2
Therefore, in spherical coordinates, a sphere of radius 2 is expressed ρ = 2.
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Although not as simple as with cylindrical coordinates, we can use spherical coordinates to describe
the equation of a cylinder.
Example 12.17: Cylinder Equation in Spherical Coordinates
Find an equation for the cylinder x 2 + y 2 = 4 in spherical coordinates.
Solution. Proceeding as in the previous example:
x 2 + y 2 = 4
ρ 2 sin 2 φ cos 2 θ + ρ 2 sin 2 φ sin 2 θ = 4
ρ 2 sin 2 φ(cos 2 θ + sin 2 θ)=4
ρ 2 sin 2 φ = 4
ρ sinφ = 2
ρ = 2
sinφ
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Exercises for 12.6
Exercise 12.6.1 Convert the following points in rectangular coordinates to cylindrical and spherical
coordinates:
(a) (1,1,1)
(b) (7,−7,5)