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Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 3 / 4B / Solution

Differentiation of

x = ρ cos ϕ

,

y = ρ sin ϕ

gives

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

The two radial face areas are

ρ_{0} Δ ϕ Δ z

and

(ρ_{0} + Δ ρ) Δ ϕ Δ z

. Each constant-

ϕ

face has area

Δ ρ Δ z

, and each constant-

z

face has area

\frac{1}{2} {(ρ_{0} + Δ ρ)^{2} - ρ_{0}^{2}} Δ ϕ .

The volume is this last expression times

Δ z

.

Only the radial faces contribute to the flux of

F (ρ) e_{ρ}

. Dividing their difference by the volume and taking

Δ ρ \to 0

gives

\frac{1}{ρ} \frac{d}{d ρ} {ρF (ρ)}

at

ρ = ρ_{0}

.

Solved by gpt-5.6-sol high.

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