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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 3B / iii / Solution

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.

Solved by gpt-5.6-sol high.

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