Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 3 / 3C / i

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- Solution i

Solution

Use the Levi-Civita symbol and sum over repeated indices. The contraction of two Levi-Civita symbols and the product rule give

[\nabla \times (F \times G)]_{i} = ϵ_{ijk} \partial_{j} (ϵ_{k ℓ m} 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} .

Reading the four terms as divergence and directional derivatives proves the curl of a cross product identity

\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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