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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/2007/paper-2/section-b/5/a/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 / 2007 / Paper 2 / Section B / 5 / a / Solution by

Codex  0 2026-10-20

u d x + v d y = d F

with continuous second partial derivatives of

F

, then

u = F_{x}

and

v = F_{y}

. The symmetry of second derivatives gives the necessary condition for an exact differential form:

(\frac{\partial u}{\partial y})_{x} = (\frac{\partial v}{\partial x})_{y} .

(1)

This condition is also locally sufficient on a small rectangle for continuously differentiable coefficients; global sufficiency additionally needs suitable topology of the domain.

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