v‾0=R−1v‾=RTv‾v‾˙=ωv‾\begin{aligned} \dot{\underline{\boldsymbol{ v}}} &= \frac{\partial \boldsymbol{ R}}{\partial t} \underline{\boldsymbol{ v}}_0 \\ &= \frac{\partial \boldsymbol{ R}}{\partial t} \boldsymbol{ R}^{\mathrm{T}}\underline{\boldsymbol{ v}}, \quad \underline{\boldsymbol{ v}}_0 = \boldsymbol{ R}^{-1}\underline{\boldsymbol{ v}} = \boldsymbol{ R}^{\mathrm{T}}\underline{\boldsymbol{ v}}\\ \dot{\underline{\boldsymbol{ v}}} &= \boldsymbol{ \omega} \underline{\boldsymbol{ v}} \end{aligned}v˙v˙=∂t∂Rv0=∂t∂RRTv,v0=R−1v=RTv=ωv The skew symmetric tensor ω is often called the spin tensor.