 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/1/a/iii/solution

Differentiate the position vector, using the angular factor from the scale factors of orthogonal coordinates:

d r = e_{ρ} d ρ + ρ e_{ϕ} d ϕ + e_{z} d z .

The basis is an orthonormal basis, so all cross terms in its squared length vanish. The Euclidean line element is therefore

∣ d r ∣^{2} = d ρ^{2} + ρ^{2} d ϕ^{2} + d z^{2} .

In particular, an angular change

d ϕ

at fixed radius traverses a physical distance

ρ ∣ d ϕ ∣

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