 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-2/6/i/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Use

w_{j} = ϵ_{j ℓ m} \partial_{ℓ} u_{m}

(w \times u)_{i} = ϵ_{ijk} ϵ_{j ℓ m} (\partial_{ℓ} u_{m}) u_{k} = u_{ℓ} \partial_{ℓ} u_{i} - u_{m} \partial_{i} u_{m} = u_{ℓ} \partial_{ℓ} u_{i} - \partial_{i} \frac{∣ u ∣ ^{2}}{2} .

This proves the convective acceleration identity, with the sign set by the displayed order

w \times u

 New to topics? Read the docs here!