 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/12/b/solution

(r, θ, z)

(ρ, ϑ, φ)

, respectively,

d V = d x d y d z, d V = r d r d θ d z, d V = ρ^{2} sin ϑ d ρ d ϑ d φ .

Here

ρ

is distance from the origin,

ϑ

is the polar angle from the positive

z

axis, and

φ

is the azimuthal angle. The spherical coordinates convention fixes which angle enters the sine factor.

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