254 CHAPTER 10. REGULAR MEASURES

10.9 Spherical Coordinates in p DimensionsSometimes there is a need to deal with spherical coordinates in more than three dimen-sions. In this section, this concept is defined and formulas are derived for these coordinatesystems. Recall polar coordinates are of the form

y1 = ρ cosθ

y2 = ρ sinθ

where ρ > 0 and θ ∈ R. Thus these transformation equations are not one to one but theyare one to one on (0,∞)× [0,2π). Here I am writing ρ in place of r to emphasize a patternwhich is about to emerge. I will consider polar coordinates as spherical coordinates intwo dimensions. I will also simply refer to such coordinate systems as polar coordinatesregardless of the dimension. This is also the reason I am writing y1 and y2 instead of themore usual x and y. Now consider what happens when you go to three dimensions. Thesituation is depicted in the following picture.

φ 1ρ

•(y1,y2,y3)

R2

R

From this picture, you see that y3 = ρ cosφ 1. Also the distance between (y1,y2) and(0,0) is ρ sin(φ 1) . Therefore, using polar coordinates to write (y1,y2) in terms of θ andthis distance,

y1 = ρ sinφ 1 cosθ ,y2 = ρ sinφ 1 sinθ ,y3 = ρ cosφ 1.

where φ 1 ∈R and the transformations are one to one if φ 1 is restricted to be in [0,π] . Whatwas done is to replace ρ with ρ sinφ 1 and then to add in y3 = ρ cosφ 1. Having done this,there is no reason to stop with three dimensions. Consider the following picture:

φ 2ρ

•(y1,y2,y3,y4)

R3

R

From this picture, you see that y4 = ρ cosφ 2. Also the distance from (y1,y2,y3) to(0,0,0) is ρ sin(φ 2) . Therefore, using polar coordinates to write (y1,y2,y3) in terms ofθ ,φ 1, and this distance,

y1 = ρ sinφ 2 sinφ 1 cosθ ,y2 = ρ sinφ 2 sinφ 1 sinθ ,y3 = ρ sinφ 2 cosφ 1,y4 = ρ cosφ 2

where φ 2 ∈ R and the transformations will be one to one if φ 2,φ 1 ∈ (0,π), θ ∈ (0,2π),ρ ∈ (0,∞) .

Continuing this way, given spherical coordinates in Rp, to get the spherical coordinatesin Rp+1, you let yp+1 = ρ cosφ p−1 and then replace every occurance of ρ with ρ sinφ p−1to obtain y1, · · · ,yp in terms of φ 1,φ 2, · · · ,φ p−1,θ , and ρ.