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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ia/paper-1/5b/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 1 / 5B / a / Solution by

Codex  0 2026-10-14

In suffix notation, repeated indices are summed from one to three. The Levi-Civita symbol and Kronecker delta contraction is

ϵ_{ijk} ϵ_{k ℓ m} = δ_{i ℓ} δ_{jm} - δ_{im} δ_{j ℓ} .

(1)

For distinct free index pairs this follows from the two possible matching permutations and their signs; if either pair repeats an index both sides vanish. Therefore

[a \times (b \times c)]_{i} = ϵ_{ijk} a_{j} ϵ_{k ℓ m} b_{ℓ} c_{m} = (δ_{i ℓ} δ_{jm} - δ_{im} δ_{j ℓ}) a_{j} b_{ℓ} c_{m} = b_{i} a_{j} c_{j} - c_{i} a_{j} b_{j} .

(2)

Recognizing the dot products proves the vector triple product identity:

a \times (b \times c) = (a \cdot c) b - (a \cdot b) c .

(3)

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