 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2011/ia/paper-2/6a/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

f

are

f_{x} = 2 x

f_{y} = 4 y (a - y^{2})

and

f_{z} = 2 z

(\frac{\partial x}{\partial y})_{z} = \frac{2 y ( y ^{2} - a )}{x}, (\frac{\partial y}{\partial z})_{x} = \frac{z}{2 y ( y ^{2} - a )}, (\frac{\partial z}{\partial x})_{y} = - \frac{x}{z} .

Their product is

- 1

wherever

x yz (y^{2} - a) \neq = 0

. The excluded factors identify exactly where these quotient formulas cannot all represent finite partial derivatives.

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