 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ia/paper-3/4b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

R^{3}

, a continuously differentiable differential one-form is exact if its mixed partial derivatives agree, equivalently if its associated vector field has zero curl. Here one can verify exactness directly by observing that

F (x, y, z) = (x^{2} + z^{2}) e^{(x + y) z}

satisfies

\frac{\partial F}{\partial x} = u, \frac{\partial F}{\partial y} = v, \frac{\partial F}{\partial z} = w .

Thus

u d x + v d y + w d z = d F

F (1, 0, 1) - F (- 1, 0, 1) = 2 e - 2 e^{- 1} .

Consequently, for every path with the stated endpoints,

\int_{(- 1, 0, 1)}^{(1, 0, 1)} (u d x + v d y + w d z) = 4 sinh 1 .

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