Past exam of the mathematical methods I course of the University of Oxford / 2018 / 2 / a

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- Solution a

Solution

Use the Levi-Civita symbol and Einstein summation convention. The tensor contraction

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

, expressed through the Kronecker delta, gives

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

The last two contractions are dot products, proving the vector triple product identity

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

.

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