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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/ia/paper-3/4a/solution

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Past exam of the mathematics course of the University of Cambridge / 2001 / ia / Paper 3 / 4A / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

Consider the globally smooth potential

Φ (x, y, z) = x y^{2} sin (x z)

. Direct differentiation gives

Φ_{x} = y^{2} sin (x z) + x y^{2} z cos (x z) = u, Φ_{y} = 2 x y sin (x z) = v, Φ_{z} = x^{2} y^{2} cos (x z) = w .

(1)

Thus the differential one-form is the exact differential

d Φ

, proving exactness rather than only checking a necessary condition on its partial derivatives. Along any piecewise differentiable path, the chain rule gives

u d x + v d y + w d z = d Φ

. The fundamental theorem of calculus therefore makes the line integral depend only on its endpoints:

\int_{(0, 0, 0)}^{(π /2, 1, 1)} (u d x + v d y + w d z) = Φ (π /2, 1, 1) - Φ (0, 0, 0) = \frac{π}{2} .

(2)

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