Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 3 / 9C / a / Solution

A Cartesian second-rank tensor has components in every orthonormal Cartesian frame that transform according towhen vector components transform as $x^{\prime }=Qx$ for an orthogonal matrix $Q$. Repeated indices are summed. If $A=B$ in one frame, their transformed difference is $Q(A-B)Q^T=0$, so their equality holds in every such frame, at the same physical point.

Now impose the scalar contraction hypothesis on $C$. For every tensor $A$, invariance in the two frames saysAt a fixed point, choose the components of $A$ to be each matrix unit in turn; each is an admissible tensor once its components in other frames are transformed. Thus every coefficient of $A_{pq}$ agrees, yielding $Q^T{C}^{\prime }Q=C$, orTherefore $C$ is necessarily a Cartesian second-rank tensor. This is the scalar contraction test for a Cartesian tensor. Testing all tensors, rather than only symmetric tensors, is essential to the proof.