 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-2/1/a/ii/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

The

y

component integrates to

ϕ = x y e^{z - x} + g (x, z)

. Matching the

x

component gives

g_{x} = 2 x

, and matching the

z

component then gives the remaining derivative zero. Consequently

ϕ (x, y, z) = x y e^{z - x} + x^{2} + C .

Direct differentiation recovers all three components, proving this is a gradient field globally on

R^{3}

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