Sometimes there is a need to deal with spherical coordinates in more than three dimensions. In this section,
this concept is defined and formulas are derived for these coordinate systems. Recall polar coordinates are
of the form
y1 = ρcosθ
y2 = ρsin θ
where ρ > 0 and θ ∈ ℝ. Thus these transformation equations are not one to one but they are one to one on
(0,∞ )
× [0,2π). Here I am writing ρ in place of r to emphasize a pattern which is about to
emerge. I will consider polar coordinates as spherical coordinates in two dimensions. I will also
simply refer to such coordinate systems as polar coordinates regardless of the dimension. This
is also the reason I am writing y_{1} and y_{2} instead of the more usual x and y. Now consider
what happens when you go to three dimensions. The situation is depicted in the following
picture.
PICT
From this picture, you see that y_{3} = ρcosϕ_{1}. Also the distance between
(y1,y2)
and
(0,0)
is ρsin
(ϕ1)
.
Therefore, using polar coordinates to write
(y1,y2)
in terms of θ and this distance,
y1 = ρsinϕ1cosθ,
y2 = ρsinϕ1sin θ,
y3 = ρcosϕ1.
where ϕ_{1}∈ ℝ and the transformations are one to one if ϕ_{1} is restricted to be in
[0,π]
. What was done is to
replace ρ with ρsinϕ_{1} and then to add in y_{3} = ρcosϕ_{1}. Having done this, there is no reason to stop with
three dimensions. Consider the following picture:
PICT
From this picture, you see that y_{4} = ρcosϕ_{2}. Also the distance between
where ϕ_{2}∈ ℝ and the transformations will be one to one if
ϕ ,ϕ ∈ (0,π),θ ∈ (0,2π),ρ ∈ (0,∞) .
2 1
Continuing this way, given spherical coordinates in ℝ^{p}, to get the spherical coordinates in ℝ^{p+1}, you let
y_{p+1} = ρcosϕ_{p−1} and then replace every occurance of ρ with ρsinϕ_{p−1} to obtain y_{1}
⋅⋅⋅
y_{p} in terms of
ϕ_{1},ϕ_{2},
⋅⋅⋅
,ϕ_{p−1},θ, and ρ.
It is always the case that ρ measures the distance from the point in ℝ^{p} to the origin in ℝ^{p}, 0. Each
ϕ_{i}∈ ℝ and the transformations will be one to one if each ϕ_{i}∈
(0,π)
, and θ ∈
(0,2π)
. Denote by
h_{p}
( ⃗ )
ρ,ϕ,θ
the above transformation.
It can be shown using math induction and geometric reasoning that these coordinates map
∏_{i=1}^{p−2}
(0,π)
×
(0,2π)
×
(0,∞)
one to one onto an open subset of ℝ^{p} which is everything except for the
set of measure zero Ψ_{p}
(N )
where N results from having some ϕ_{i} equal to 0 or π or for ρ = 0 or for θ equal
to either 2π or 0. Each of these are sets of Lebesgue measure zero and so their union is also a set of
measure zero. You can see that h_{p}
(∏ )
pi−=21 (0,π)× (0,2π )× (0,∞ )
omits the union of the coordinate axes
except for maybe one of them. This is not important to the integral because it is just a set of measure
zero.
Theorem 8.5.1Let y = h_{p}
( )
⃗ϕ,θ,ρ
be the spherical coordinate transformations in ℝ^{p}. Then lettingA = ∏_{i=1}^{p−2}
(0,π)
×
(0,2π)
, it follows h maps A ×
(0,∞ )
one to one onto all of ℝ^{p}except a set ofmeasure zero given by h_{p}
and θ.^{1}Then if f is nonnegative and Lebesgue measurable,
∫ ∫ ∫ ( ( )) ( )
f (y) dmp = f (y )dmp = f hp ⃗ϕ,θ,ρ ρp−1Φ ⃗ϕ,θ dmp (8.12)
ℝp hp(A) A
(8.12)
Furthermore whenever f is Borel measurable and nonnegative, one can apply Fubini’s theorem andwrite
∫ ∫ ∫
∞ p−1 ( (⃗ )) (⃗ ) ⃗
ℝp f (y)dy = 0 ρ A f h ϕ,θ,ρ Φ ϕ,θ dϕd θdρ (8.13)
(8.13)
where here d
⃗ϕ
dθ denotes dm_{p−1}on A. The same formulas hold if f ∈ L^{1}
(ℝp)
.
Proof: Formula 8.11 is obvious from the definition of the spherical coordinates because in the matrix of
the derivative, there will be a ρ in p− 1 columns. The first claim is also clear from the definition and math
induction or from the geometry of the above description. It remains to verify 8.12 and 8.13. It is clear h_{p}
maps A× [0,∞) onto ℝ^{p}. Since h_{p} is differentiable, it maps sets of measure zero to sets of measure zero.
Then
p
ℝ = hp(N ∪ A × (0,∞ )) = hp (N )∪ hp(A × (0,∞ )),
the union of a set of measure zero with h_{p}
(A × (0,∞ ))
. Therefore, from the change of variables
formula,
∫ ∫ ∫ ( ( )) ( )
p f (y)dmp = f (y)dmp = f hp ⃗ϕ,θ,ρ ρp−1Φ ⃗ϕ,θ dmp
ℝ hp(A ×(0,∞)) A× (0,∞ )
which proves 8.12. This formula continues to hold if f is in L^{1}
p
(ℝ )
. Finally, if f ≥ 0 or in L^{1}
n
(ℝ )
and is
Borel measurable, then it is ℱ^{p} measurable as well. Recall that ℱ^{p} includes the smallest σ algebra which
contains products of open intervals. Hence ℱ^{p} includes the Borel sets ℬ
p
(ℝ )
. Thus from the definition of
m_{p}
∫ ( ( )) p−1 ( )
f hp ⃗ϕ,θ,ρ ρ Φ ⃗ϕ,θ dmp
A× (0,∞ )
∫ ∫ ( ( )) ( )
= f hp ⃗ϕ,θ,ρ ρp− 1Φ ⃗ϕ,θ dmp−1dm
∫(0,∞ ) A ∫ ( ( )) ( )
= ρp− 1 f hp ⃗ϕ,θ,ρ Φ ⃗ϕ,θ dmp −1dm
(0,∞ ) A
Now the claim about f ∈ L^{1} follows routinely from considering the positive and negative parts of the real
and imaginary parts of f in the usual way. ■
Note that the above equals
∫ ( ( )) ( )
f hp ⃗ϕ,θ,ρ ρp−1Φ ⃗ϕ,θ dmp
A¯× [0,∞ )
and the iterated integral is also equal to
∫ ∫ ( ( )) ( )
ρp−1 f hp ⃗ϕ,θ,ρ Φ ⃗ϕ,θ dmp −1dm
[0,∞ ) ¯A
because the difference is just a set of measure zero.
Notation 8.5.2Often this is written differently. Note that from the spherical coordinate formulas,f
( ( ))
h ⃗ϕ,θ,ρ
= f
(ρω )
where
|ω |
= 1. Letting S^{p−1}denote the unit sphere,
{ω ∈ ℝp : |ω | = 1}
, theinside integral in the above formula is sometimes written as
∫
f (ρω)dσ
Sp−1
where σ is a measure on S^{p−1}. See [80]for another description of this measure. It isn’t an important issuehere. Either 8.13or the formula
∫ ∞ ( ∫ )
ρp−1 f (ρω)dσ dρ
0 Sp−1
will be referred to as polar coordinates and is very useful in establishing estimates. Hereσ
( )
Sp−1
≡∫_{A}Φ
( )
⃗ϕ,θ
dm_{p−1}.
Example 8.5.3For what values of s is the integral∫_{B}
(0,R)
( )
1 + |x|2
^{s}dy bounded independent ofR? Here B
(0,R )
is the ball,
p
{x ∈ ℝ : |x| ≤ R}
.
I think you can see immediately that s must be negative but exactly how negative? It turns out it
depends on p and using polar coordinates, you can find just exactly what is needed. From the polar
coordinates formula above,