Velocity gradient ($\nabla \mathbf{u}$)

The velocity gradient has components $(\nabla \mathbf{u}{)}_{ij}={\partial }_j{u}_i$. Its symmetric part is the deformation-rate tensor and its antisymmetric part is the local spin.

The velocity divergence is the trace of the velocity gradient. It is the local fractional rate of volume expansion of a moving material element.