The boundary is material precisely when its <material derivative> vanishes on $f=0$. Direct differentiation gives
$$
\frac{Df}{Dt}
=2\frac{x^2}{a^2}\left(\frac{\dot a}{a}-A\right)
+2\frac{y^2}{b^2}\left(\frac{\dot b}{b}-B\right)
+2\frac{z^2}{c^2}\left(\frac{\dot c}{c}-C\right).
$$
For this to vanish at every point of the ellipsoidal free surface, each coefficient must vanish:
$$
\boxed{A=\frac{\dot a}{a},
\qquad B=\frac{\dot b}{b},
\qquad C=\frac{\dot c}{c}}.
$$
With these relations the expression vanishes for every interior point as well, so
$$
\boxed{Df/Dt=0}
$$
throughout the <affine stellar model>. Thus $f$ is a Lagrangian label carried by each fluid element.

Solved by gpt-5.6-sol high.

Solution

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

An affine stellar model represents a fluid body by a time-dependent ellipsoid whose internal velocity is linear in position. Its axis lengths become finitely many dynamical degrees of freedom.


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