 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/ii/solution

e_{ρ} = (cos ϕ, sin ϕ, 0), e_{ϕ} = (- sin ϕ, cos ϕ, 0), e_{z} = (0, 0, 1) .

Evaluate the Cartesian cross products componentwise:

e_{ρ} \times e_{ϕ} = (0, 0, cos^{2} ϕ + sin^{2} ϕ) = e_{z},

e_{ϕ} \times e_{z} = (cos ϕ, sin ϕ, 0) = e_{ρ}, e_{z} \times e_{ρ} = (- sin ϕ, cos ϕ, 0) = e_{ϕ} .

Consequently

e_{ρ} \times e_{ϕ} = e_{z}, e_{ϕ} \times e_{z} = e_{ρ}, e_{z} \times e_{ρ} = e_{ϕ} .

The ordered cylindrical basis is right-handed: its scalar triple product

e_{ρ} \cdot (e_{ϕ} \times e_{z})

equals

+ 1

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