 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ib/paper-1/17c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

Let

ω = \nabla \times u

be the vorticity. In component notation, contracting two Levi-Civita symbols gives

(u \times ω)_{i} = ε_{ijk} u_{j} ε_{k l m} \partial_{l} u_{m} = u_{j} \partial_{i} u_{j} - u_{j} \partial_{j} u_{i} .

The first term is

\partial_{i} (∣ u ∣^{2} /2)

and the second is the

i

th component of the convective acceleration. Rearranging proves the velocity identity

(u \cdot \nabla) u = \nabla (\frac{1}{2} ∣ u ∣^{2}) - u \times ω .

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