Page 98 - Vector Analysis

94 CHAPTER 3. Multiple Integrals

Example 3.37 (Cylindrical coordinates). In R3, when the domain over which the integral
is taken is a cylinder C; that is, C = D ˆ [a, b] for some disk D and ´8 ă a ă b ă R, then
the change of variables

ψ(r, θ, z) = (x0 + r cos θ, y0 + r sin θ, z) 0 ă r ă R , 0 ă θ ă 2π , a ď z ď b ,

where (x0, y0) is the center of D and R is the radisu of D, is sometimes very useful for
evaluating the integral. Since the Jacobian of ψ is

ˇ B ψ1 B ψ1 B ψ1 ˇ
ˇ ˇ

ˇ Br Bθ Bz ˇ ˇˇcos θ ´r sin θ 0ˇˇ
ˇ ˇ

Jψ (r, θ, z) = ˇ B ψ2 B ψ2 B ψ2 ˇ = ˇ θ r cos θ ˇ
ˇ Br Bθ ˇ ˇsin 0ˇ = r ,
ˇ ˇ
ˇ Bz ˇ ˇ
ˇ ˇ ˇ

ˇ B ψ3 B ψ3 B ψ3 ˇ ˇ0 0 1ˇ
ˇ ˇ

ˇ Br Bθ Bz ˇ

we must have

żż

f (x, y, z) d(x, y, z) = f (x, y, z) d(x, y, z)

C ψ((0,R)ˆ(0,2π)ˆ[a,b])

ż
= (f ˝ ψ)(r, θ, z)ˇˇJψ(r, θ, z)ˇˇ d(r, θ, z)

(0,R)ˆ(0,2π)ˆ[a,b]

ż

= f (x0 + r cos θ, y0 + r sin θ, z) r d(r, θ, z) .

(0,R)ˆ(0,2π)ˆ[a,b]

Example 3.38 (Spherical coordinates). In R3, when the domain over which the integral is
taken is a ball B, the change of variables

ψ(ρ, θ, ϕ) = (x0 + ρ cos θ sin ϕ, y0 + ρ sin θ sin ϕ, z0 + ρ cos ϕ) 0 ă ρ ă R, 0 ă θ ă 2π, 0 ă ϕ ă π,

where (x0, y0, z0) is the center of B and R is the radius of B, is often used to evaluate the
integral a function over B. Since the Jacobian of ψ is

ˇ B ψ1 B ψ1 B ψ1 ˇ
ˇ ˇ

ˇ B ρ Bθ Bϕ ˇ ˇcos θ sin ϕ ´ρ sin θ sin ϕ ρ cos θ cos ϕˇˇ
ˇ ˇ ˇ

Jψ (ρ, θ, ϕ) = ˇ B ψ2 B ψ2 B ψ2 ˇ = ˇ θ sin ϕ ρ cos θ sin ϕ ˇ
ˇ Bρ Bθ ˇ ˇsin ρ sin θ cos ϕˇ
ˇ ˇ
ˇ Bϕ ˇ ˇ ˇ

ˇˇ
ˇ cos ϕ 0 ´ρ sin ϕ ˇ
ˇˇ
ˇ B ψ3 B ψ3 B ψ3 ˇ
ˇˇ

ˇ Bρ Bθ Bϕ ˇ

= ´ρ2 cos2 θ sin3 ϕ ´ ρ2 sin2 θ sin ϕ cos2 ϕ ´ ρ2 cos2 θ sin ϕ cos2 ϕ ´ ρ2 sin2 θ sin3 ϕ

= ´ρ2 sin3 ϕ ´ ρ2 sin ϕ cos2 ϕ = ´ρ2 sin ϕ ,