 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-2/12/b/solution

Hold

r

fixed and differentiate the position vector

X (θ, ϕ)

. Its coordinate tangent vectors are

X_{θ} = r (cos θ cos ϕ, cos θ sin ϕ, - sin θ), X_{ϕ} = r (- sin θ sin ϕ, sin θ cos ϕ, 0) .

Their dot product is zero, and their squared Euclidean norms are

r^{2}

and

r^{2} sin^{2} θ

. To first order, the displacement is

δ X = X_{θ} δ θ + X_{ϕ} δ ϕ + O ((∣ δ θ ∣ + ∣ δ ϕ ∣)^{2})

. Squaring gives

δ s^{2} ≃ r^{2} (δ θ^{2} + sin^{2} θ δ ϕ^{2}) .

This is the line element on a sphere. The chord distance and the short surface arc length agree to the order retained here.

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