(a) Define a rank two tensor and show that if two rank two tensors and are the same in one Cartesian coordinate system, then they are the same in all Cartesian coordinate systems.
The quantity has the property that, for every rank two tensor , the quantity is a scalar. Is necessarily a rank two tensor? Justify your answer with a proof from first principles, or give a counterexample.
(b) Show that, if a tensor is invariant under rotations about the -axis, then it has the form
(c) The inertia tensor about the origin of a rigid body occupying volume and with variable mass density is defined to be
The rigid body has uniform density and occupies the cylinder
Show that the inertia tensor of about the origin is diagonal in the coordinate system, and calculate its diagonal elements.