 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ia/paper-4/11b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Let

e_{1}^{'} (t), e_{2}^{'} (t), e_{3}^{'} (t)

S^{'}

. A small rotation through

ω d t

changes each basis vector by

ω \times e_{j}^{'} d t

(\frac{d e _{j}^{'}}{d t})_{S} = ω \times e_{j}^{'} .

Write the arbitrary vector as

a = \sum_{j} a_{j} (t) e_{j}^{'} (t)

. Differentiating the components and the moving basis separately gives

(\frac{d a}{d t})_{S} = \sum_{j} \overset{a}{˙}_{j} e_{j}^{'} + \sum_{j} a_{j} ω \times e_{j}^{'} = (\frac{d a}{d t})_{S^{'}} + ω \times a .

This proves the rotating-frame derivative formula, including when the angular velocity varies with time.

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