Paper 3, Section II, B

Vector Calculus | Part IA, 2021

(a) By a suitable change of variables, calculate the volume enclosed by the ellipsoid x2/a2+y2/b2+z2/c2=1x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1, where a,ba, b, and cc are constants.

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

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

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

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

(c) A half-ellipsoidal membrane S\mathcal{S} is described by the open surface 4x2+4y2+z2=44 x^{2}+4 y^{2}+z^{2}=4, with z0z \geqslant 0. At a given instant, air flows beneath the membrane with velocity u=\mathbf{u}= (y,x,α)(-y, x, \alpha), where α\alpha is a constant. The flow exerts a force on the membrane given by

Fi=Sβ2uiujnjdSF_{i}=\int_{\mathcal{S}} \beta^{2} u_{i} u_{j} n_{j} d S

where β\beta is a constant parameter.

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

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

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