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

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- Solution ii

Solution

Dividing the coordinate tangents by their scale factors of orthogonal coordinates gives an orthonormal basis

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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