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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-2/8/c/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 / 8 / c / Solution by

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

The origin is excluded by the denominator. On the punctured plane, direct differentiation gives

d (- \frac{x}{x ^{2} + y ^{2}}) = \frac{( x ^{2} - y ^{2} ) d x + 2 x y d y}{( x ^{2} + y ^{2} ) ^{2}} .

(1)

Hence the form is exact on its whole stated domain, with

f = - x / (x^{2} + y^{2}) + C

. Although the punctured plane is not simply connected, the explicitly constructed single-valued potential establishes global exactness directly; a topological sufficiency criterion is unnecessary here.

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