Paper 3, Section II, B

(a) By a suitable change of variables, calculate the volume enclosed by the ellipsoid $x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1$, where $a, b$, and $c$ are constants.

(b) Suppose $T_{i j}$ is a second rank tensor. Use the divergence theorem to show that

$\int_{\mathcal{S}} T_{i j} n_{j} d S=\int_{\mathcal{V}} \frac{\partial T_{i j}}{\partial x_{j}} d V$

where $\mathcal{S}$ is a closed surface, with unit normal $n_{j}$, and $\mathcal{V}$ is the volume it encloses.

[Hint: Consider $e_{i} T_{i j}$ for a constant vector $\left.e_{i} .\right]$

(c) A half-ellipsoidal membrane $\mathcal{S}$ is described by the open surface $4 x^{2}+4 y^{2}+z^{2}=4$, with $z \geqslant 0$. At a given instant, air flows beneath the membrane with velocity $\mathbf{u}=$ $(-y, x, \alpha)$, where $\alpha$ is a constant. The flow exerts a force on the membrane given by

$F_{i}=\int_{\mathcal{S}} \beta^{2} u_{i} u_{j} n_{j} d S$

where $\beta$ is a constant parameter.

Show the vector $a_{i}=\partial\left(u_{i} u_{j}\right) / \partial x_{j}$ can be rewritten as $\mathbf{a}=-(x, y, 0)$.

Hence use $(*)$ to calculate the force $F_{i}$ on the membrane.