 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/ia/paper-3/12c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-07

Use the Levi-Civita symbol and summation over repeated indices. Contracting the two symbols gives

[\nabla \times (F \times G)]_{i} = ϵ_{ijk} \partial_{j} (ϵ_{k l m} F_{l} G_{m}) = (δ_{i l} δ_{jm} - δ_{im} δ_{j l}) \partial_{j} (F_{l} G_{m}) = \partial_{j} (F_{i} G_{j} - F_{j} G_{i}) = F_{i} \partial_{j} G_{j} - G_{i} \partial_{j} F_{j} + G_{j} \partial_{j} F_{i} - F_{j} \partial_{j} G_{i} .

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

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