 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/6/vii/solution

Write the curl and cross product with the Levi-Civita symbol, then use its contraction identity:

[\nabla \times (A \times B)]_{i} = ϵ_{ijk} \partial_{j} (ϵ_{k ℓ m} A_{ℓ} B_{m}) = (δ_{i ℓ} δ_{jm} - δ_{im} δ_{j ℓ}) \partial_{j} (A_{ℓ} B_{m}) = \partial_{j} (A_{i} B_{j} - A_{j} B_{i}) .

Expanding with the product rule gives

A_{i} \partial_{j} B_{j} + B_{j} \partial_{j} A_{i} - B_{i} \partial_{j} A_{j} - A_{j} \partial_{j} B_{i} .

Recognize the divergence in the first and third terms, and the directional derivatives in the other two. The curl of a cross product identity is therefore

\nabla \times (A \times B) = A (\nabla \cdot B) + (B \cdot \nabla) A - B (\nabla \cdot A) - (A \cdot \nabla) B .

The derivatives act on both vector fields. No assumption of vanishing divergence or spatially constant fields has been made.

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