Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 3 / a / Solution

The antisymmetric velocity-gradient tensor $\mathbf{\Omega }$ is the local angular velocity of a rigid-body rotation. Under such a rotation a material vector must rotate at exactly the same angular velocity, independently of its molecular constitution. The coefficient of $\mathbf{\Omega }\cdot \mathbf{p}$ is therefore fixed to one by rotational covariance, also called material objectivity. In contrast, the symmetric strain-rate tensor $\mathbf{D}$ deforms rather than merely rotates the material, so its flow-alignment coefficient $\xi$ is material dependent.