20.3. CYLINDRICAL AND SPHERICAL COORDINATES 375

Example 20.3.8 Find the volume element for cylindrical coordinates.

In cylindrical coordinates,  xyz

=

 r cosθ

r sinθ

z

Therefore, the Jacobian determinant is

det

 cosθ −r sinθ 0sinθ r cosθ 0

0 0 1

= r.

It follows the volume element in cylindrical coordinates is r dθ dr dz.

Example 20.3.9 In the cone of Example 20.3.7 set up the integrals for finding the volumein cylindrical coordinates.

This is a better coordinate system for this example than spherical coordinates. Thistime you should get ∫ 2π

0

∫ 4

0

∫ 4

rrdzdrdθ =

643

π

Example 20.3.10 This example uses spherical coordinates to verify an important conclu-sion about gravitational force. Let the hollow sphere, H be defined by a2 < x2+y2+z2 < b2

and suppose this hollow sphere has constant density taken to equal 1. Now place a unitmass at the point (0,0,z0) where |z0| ∈ [a,b] . Show that the force of gravity acting on this

unit mass is(

αG∫

H(z−z0)

[x2+y2+(z−z0)2]

3/2 dV)k and then show that if |z0| > b then the force

of gravity acting on this point mass is the same as if the entire mass of the hollow spherewere placed at the origin, while if |z0| < a, the total force acting on the point mass fromgravity equals zero. Here G is the gravitation constant and α is the density. In particular,this shows that the force a planet exerts on an object is as though the entire mass of theplanet were situated at its center1.

Without loss of generality, assume z0 > 0. Let dV be a little chunk of material locatedat the point (x,y,z) of H the hollow sphere. Then according to Newton’s law of gravity, theforce this small chunk of material exerts on the given point mass equals

xi+ yj+(z− z0)k

|xi+ yj+(z− z0)k|1(

x2 + y2 +(z− z0)2)Gα dV =

1This was shown by Newton in 1685 and allowed him to assert his law of gravitation applied to the planets asthough they were point masses. It was a major accomplishment.