Paper 3, Section II, C

Vector Calculus | Part IA, 2010

(a) Define a rank two tensor and show that if two rank two tensors AijA_{i j} and BijB_{i j} are the same in one Cartesian coordinate system, then they are the same in all Cartesian coordinate systems.

The quantity CijC_{i j} has the property that, for every rank two tensor AijA_{i j}, the quantity CijAijC_{i j} A_{i j} is a scalar. Is CijC_{i j} necessarily a rank two tensor? Justify your answer with a proof from first principles, or give a counterexample.

(b) Show that, if a tensor TijT_{i j} is invariant under rotations about the x3x_{3}-axis, then it has the form

(αω0ωα000β)\left(\begin{array}{ccc} \alpha & \omega & 0 \\ -\omega & \alpha & 0 \\ 0 & 0 & \beta \end{array}\right)

(c) The inertia tensor about the origin of a rigid body occupying volume VV and with variable mass density ρ(x)\rho(\mathbf{x}) is defined to be

Iij=Vρ(x)(xkxkδijxixj)dVI_{i j}=\int_{V} \rho(\mathbf{x})\left(x_{k} x_{k} \delta_{i j}-x_{i} x_{j}\right) \mathrm{d} V

The rigid body BB has uniform density ρ\rho and occupies the cylinder

{(x1,x2,x3):2x32,x12+x221}\left\{\left(x_{1}, x_{2}, x_{3}\right):-2 \leqslant x_{3} \leqslant 2, x_{1}^{2}+x_{2}^{2} \leqslant 1\right\}

Show that the inertia tensor of BB about the origin is diagonal in the (x1,x2,x3)\left(x_{1}, x_{2}, x_{3}\right) coordinate system, and calculate its diagonal elements.

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