Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 314 / 1 / a

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- Solution a

Solution

The boundary is material precisely when its material derivative vanishes on

f = 0

. Direct differentiation gives

\frac{D f}{D t} = 2 \frac{x ^{2}}{a ^{2}} (\frac{a ˙}{a} - A) + 2 \frac{y ^{2}}{b ^{2}} (\frac{b ˙}{b} - B) + 2 \frac{z ^{2}}{c ^{2}} (\frac{c ˙}{c} - C) .

For this to vanish at every point of the ellipsoidal free surface, each coefficient must vanish:

A = \frac{a ˙}{a}, B = \frac{b ˙}{b}, C = \frac{c ˙}{c} .

With these relations the expression vanishes for every interior point as well, so

D f / D t = 0

throughout the affine stellar model. Thus

f

is a Lagrangian label carried by each fluid element.

Solved by gpt-5.6-sol high.

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