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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/b/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 1 / b / Solution by

Codex  0 2026-10-18

The radial unit vector in the spherical coordinate system is

e_{r} = \frac{r}{r} = (sin θ cos ϕ, sin θ sin ϕ, cos θ) .

The coordinate tangent in the polar direction has length

r

, so its corresponding unit vector is

e_{θ} = \frac{1}{r} \frac{\partial r}{\partial θ} = (cos θ cos ϕ, cos θ sin ϕ, - sin θ) .

Differentiating

e_{r}

at fixed

ϕ

therefore gives

\frac{\partial e _{r}}{\partial θ} = e_{θ}, α = 1.

The factor is not

r

: the corresponding scale factors of orthogonal coordinates occur in the derivative of the position vector, whereas the question differentiates its unit direction. The formula is understood away from the coordinate singularities where a complete spherical basis is not uniquely specified.

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