Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 12 / ii

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- Solution ii

Solution

Apply the contraction of two Levi-Civita symbols to the cross product:

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

Thus the vector triple product identity is

A \times (B \times C) = B (A \cdot C) - C (A \cdot B) .

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