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Past exam of the mathematics course of the University of Cambridge

2024

Paper 3

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
$$
G=e_x,
\qquad F=ye_z.
$$
Then $\nabla\times G=0$, so the proposed left side is zero, whereas
$$
G\times F=-ye_y,
\qquad
\nabla\cdot(G\times F)=-1.
$$
\b[Thus the identity is false.]


Solution

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

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

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

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