 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-4/16c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

S (x, y, z, t) = z - η (x, y, t) = 0.

D S / D t = 0

. Thus, on

z = η

0 = S_{t} + u S_{x} + v S_{y} + w S_{z} = - η_{t} - u η_{x} - v η_{y} + w .

Therefore

w = η_{t} + u η_{x} + v η_{y} = \frac{Dη}{D t}

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