354 CHAPTER 17. SOME PHYSICAL APPLICATIONS

when φ is given, the location of the point is narrowed down to a circle of “latitude” andfinally, θ determines which point is on this circle by specifying a circle of “longitude”. Letφ ∈ [0,π],θ ∈ [0,2π), and ρ ∈ [0,∞). The picture shows how to relate these new coordinatesystems to Cartesian coordinates. Note that θ is the same in the two coordinate systemsand that ρ sinφ = r.

If you fix two of the variables and take a derivative with respect to the other, you arefinding the tangent vector to a space curve. There are three of these space curves

ρ →

 ρ sin(φ)cos(θ)ρ sin(φ)sin(θ)

ρ cos(φ)

 ,θ →

 ρ sin(φ)cos(θ)ρ sin(φ)sin(θ)

ρ cos(φ)

 ,φ →

 ρ sin(φ)cos(θ)ρ sin(φ)sin(θ)

ρ cos(φ)

Doing derivatives with respect to ρ,θ , and φ for these three space curves gives tangentvectors, the first in the direction of increasing ρ for fixed θ ,φ , the second in the directionof increasing θ fixing ρ,φ , and the third in the direction of increasing φ for fixed θ ,ρ .These tangent vectors are sin(φ)cos(θ)

sin(φ)sin(θ)cos(φ)

 ,

 −ρ sinθ sinφ

ρ cosθ sinφ

0

 ,

 ρ cosθ cosφ

ρ cosφ sinθ

−ρ sinφ

You should note that the dot product of any two different vectors is zero. This is why thesespherical coordinates are known as an orthogonal. This procedure is important for generalcurvilinear coordinates.

It is often convenient to divide these vectors by their lengths to obtain unit vectors inthe given directions. Also it is very useful to list them in an order that the vectors are a righthanded system. I will do this later by listing them according to differentiating with respectto φ first, then with respect to θ and then with respect to ρ . This yields the following unitvectors in this order: cosθ cosφ

cosφ sinθ

−sinφ

 ,

 −sinθ

cosθ

0

 ,

 sin(φ)cos(θ)sin(φ)sin(θ)

cos(φ)

You could denote these vectors respectively as i,j,k and from the geometrical definitionof the cross product, it follows that i×j = k j×k= i, etc. Here is a picture illustratingthese vectors in the order just described at a point (ρ,θ ,φ) of the sphere of radius ρ . Thefirst is tangent to a line of longitude, the second, a line of latitude and the third pointsdirectly out away from the sphere of radius ρ .

i

kj

17.2 Exercises1. The following are the polar coordinates of points. Find the rectangular coordinates.