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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/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 / 2011 / Paper 1 / 18 / a / Solution by

Codex  0 2026-10-18

If the form is

d F

for a scalar function with continuous second partial derivatives, then

p = F_{x}

and

q = F_{y}

. Equality of the mixed partial derivatives therefore requires

\frac{\partial p}{\partial y} = \frac{\partial q}{\partial x} .

(1)

This is necessary for an exact differential. For continuously differentiable coefficients on a simply connected open domain it is also sufficient; on a domain with holes, the equality alone need not give a single-valued global potential.

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