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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/i/solution

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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 / i / Solution by

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

A differentiable gradient field must have equal mixed derivatives. Here

\partial F_{y} / \partial z = 3 z^{2}

but

\partial F_{z} / \partial y = - x

, which are not identically equal on any open region. Equivalently,

\nabla \times F = (- x - 3 z^{2}, x + y - 3 z^{2}, 0) .

(1)

Thus no scalar potential exists for this field on an open region. Vanishing of the curl on a special lower-dimensional set would not supply a three-dimensional potential.

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