
6-6 Calculus – Differentiation and Integration
Conversion Between Spherical and Cylindrical Coordinates
We can convert spherical coordinates into cylindrical coordinates using the following formulae:
ρ θ ρ θ ϕ ϕ θ θ= = = = =
r r z r z rcos , sin , , cos sin .and
Cylindrical coordinates can be converted into spherical coordinates using the following formulae:
r z
z
r
z
r
= + =
=
=
− −
ρ θ θ ϕ ϕ
2 2 1 1
, sin , cos .and
Example 6.6 Convert the cylindrical coordinates
6
6
2, ,
p
to spherical coordinates.
Solution: We have
r z= +r
2 2
, q = sin
-1
(z/r), q = cos
-1
(z/r) and f = f.
Thus, r = + = =6 2 8 2 2.
And, cos ( / )
cos
cos
θ
π
=
=