Rotation tensor

A rotation tensor R\boldsymbol{ R} has the property that RT=R−1\boldsymbol{ R}^{\mathrm{T}}=\boldsymbol{ R}^{-1}, and rotates a vector v‾\underline{\boldsymbol{ v}} by right-multiplication: v‾′=Rv‾\underline{\boldsymbol{ v}}' = \boldsymbol{ R}\underline{\boldsymbol{ v}}.

Coordinate transformation tensor

Given the basis system e‾i\underline{\boldsymbol{ e}}_{ i} and a rotated basis system g‾i=Re‾i\underline{\boldsymbol{ g}}_i = \boldsymbol{ R}\underline{\boldsymbol{ e}}_{ i}, where R=Rije‾i⊗e‾j\boldsymbol{ R}=R_{ij}\underline{\boldsymbol{ e}}_{ i}\otimes\underline{\boldsymbol{ e}}_{ j}, the coordinate transformation tensor from e‾i\underline{\boldsymbol{ e}}_{ i} to g‾i\underline{\boldsymbol{ g}}_i is

Q=(g‾i⋅e‾j)g‾i⊗e‾j=RijTg‾i⊗e‾j=I\begin{aligned} \boldsymbol{ Q} = (\underline{\boldsymbol{ g}}_i \cdot \underline{\boldsymbol{ e}}_j) \underline{\boldsymbol{ g}}_i \otimes \underline{\boldsymbol{ e}}_j = R_{ij}^{\mathrm{T}} \underline{\boldsymbol{ g}}_i \otimes \underline{\boldsymbol{ e}}_j = \boldsymbol{ I} \end{aligned}

How to rotate/change basis system

Single contraction from the left of each base vector with the rotation/coordinate transformation tensor. In practice, left contraction with each free index.

Differentiation

Consider a rotation as a function of time, i.e. R(t)\boldsymbol{ R}(t). Then

∂RTR∂t=0=∂RT∂tR+RT∂R∂t[RT∂R∂t]T=−RT∂R∂t⇒Skew symmetric\begin{aligned} \frac{\partial \boldsymbol{ R}^{\mathrm{T}}\boldsymbol{ R}}{\partial t} = \boldsymbol{ 0} &= \frac{\partial \boldsymbol{ R}^{\mathrm{T}}}{\partial t}\boldsymbol{ R} + \boldsymbol{ R}^{\mathrm{T}}\frac{\partial \boldsymbol{ R}}{\partial t} \\ \left[\boldsymbol{ R}^{\mathrm{T}}\frac{\partial \boldsymbol{ R}}{\partial t}\right]^{\mathrm{T}} &= -\boldsymbol{ R}^{\mathrm{T}}\frac{\partial \boldsymbol{ R}}{\partial t} \Rightarrow \text{Skew symmetric} \end{aligned}

If we now consider rotation of vector v‾=R(t)⋅v‾0\underline{\boldsymbol{ v}}=\boldsymbol{ R}(t) \cdot \underline{\boldsymbol{ v}}_0 with time, starting at v‾0\underline{\boldsymbol{ v}}_0, we have

v‾˙=∂R∂tv‾0=∂R∂tRTv‾,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}

The skew symmetric tensor ω\boldsymbol{ \omega} is often called the spin tensor.

© Knut Andreas Meyer.
Last modified: October 31, 2025.
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