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 Incoming links: Second vertical moment of a solid ring torus

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 1 / c / Solution 2026-10-18

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Only the quartic term contributes to the Laplacian. The mixed cubic has zero second derivative in each coordinate, while the exponential–trigonometric pairs are harmonic:

Δ (z^{4}) = 12 z^{2}, Δ (x yz) = 0, Δ (e^{- 3 y} cos 3 x) = 0, Δ (e^{2 x} sin 2 y) = 0.

(1)

Using

z = r cos α

and the Jacobian determinant from part (a), the divergence theorem gives

I = 12 \int_{0}^{2 π} \int_{- π}^{π} \int_{0}^{R} r^{3} cos^{2} α (1 + r sin α) d r d α d β = 12 (2 π) (\int_{- π}^{π} cos^{2} α d α) \frac{R ^{4}}{4} = 6 π^{2} R^{4} .

(2)

The term containing

sin α cos^{2} α

integrates to zero. Equivalently, the second vertical moment of a solid ring torus with major radius one is

\int_{V} z^{2} d V = π^{2} R^{4} /2

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