\[\text{Rate of angular deformation in } xy \text{ plane} =\left(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\right)\]

\[\text{Rate of angular deformation in } yz \text{ plane} =\left(\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}\right)\]

\[\text{Rate of angular deformation in } zx \text{ plane}= \left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)\]

Rate of rotation (angular velocity) at a point is defined as the average rotation rate of two initially perpendicular lines that intersect at that point

Rate of translation vector in Cartesian coordinates \[ \vec{V} = u\,\vec{i} + v\,\vec{j} + w\,\vec{k} \]

Rate of rotation vector in Cartesian coordinates \[\vec{\omega}= \frac{1}{2} \left(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}\right)\vec{i} + \frac{1}{2} \left(\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}\right)\vec{j} + \frac{1}{2} \left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)\vec{k} \]