 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ia/paper-3/3b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

The first identity in part (ii), with

F

and

G

interchanged, says

\nabla \cdot (G \times F) + = F \cdot (\nabla \times G) - G \cdot (\nabla \times F),

so the proposed identity is missing its final term. For a concrete counterexample, take

G = e_{x}, F = y e_{z} .

Then

\nabla \times G = 0

, so the proposed left side is zero, whereas

G \times F = - y e_{y}, \nabla \cdot (G \times F) = - 1.

Thus the identity is false.

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