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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/2011/paper-1/18/b/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 / 2011 / Paper 1 / 18 / b / i / Solution by

Codex  0 2026-10-18

For the printed coefficients,

p = 2 sin x cosh y

and

q = sin^{2} x sinh y

. Their cross derivatives are

p_{y} = 2 sin x sinh y, q_{x} = 2 sin x cos x sinh y .

(1)

They are not equal as functions; their difference is

2 sin x (1 - cos x) sinh y

. Hence the form is not exact. Agreement on certain lines or isolated points is insufficient, because exactness requires the derivative identity throughout an open domain.

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