The body angular velocity of a $C^1$ rotation path: $\mathrm{\Omega }=R^{\mathsf{T}}{R}^{\mathrm{\prime }}$ is skew, $\mathrm{\Omega }z=\omega \times z$ determines a unique continuous $\omega$, and $R^{\mathrm{\prime }}=R\mathrm{\Omega }$

Let $I\subseteq \mathbb{R}$ be a nondegenerate open interval, let $SO(3)$ be as in The orthogonal group $O(3)$ and the rotation group $SO(3)$, and let $R:I\to SO(3)$ be of class $C^1$, that is, all nine entries of $R$ have continuous first derivatives; write $R^{\mathrm{\prime }}(t)$ for the entrywise derivative and put

$\mathrm{\Omega }(t):=R(t{)}^{\mathsf{T}}{R}^{\mathrm{\prime }}(t)\in M_3(\mathbb{R})$

for the product of the transpose of $R(t)$ with $R^{\mathrm{\prime }}(t)$ (Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes, The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations). Then:

1. $\mathrm{\Omega }(t)$ is skew-symmetric for every $t\in I$, that is $\mathrm{\Omega }(t{)}^{\mathsf{T}}=-\mathrm{\Omega }(t)$.
2. There is exactly one map $\omega :I\to {\mathbb{R}}^3$ with

$R(t{)}^{\mathsf{T}}{R}^{\mathrm{\prime }}(t)z=\omega (t)\times z(z\in {\mathbb{R}}^3,t\in I),$

namely $\omega (t)=(-{\mathrm{\Omega }}_{23}(t),{\mathrm{\Omega }}_{13}(t),-{\mathrm{\Omega }}_{12}(t))$, and this $\omega$ is continuous. It is called the body angular velocity of $R$. 3. For every $t\in I$, writing ${\mathrm{\Omega }}_{\omega (t)}$ for the matrix $z\mapsto \omega (t)\times z$, one has $R^{\mathrm{\prime }}(t)=R(t){\mathrm{\Omega }}_{\omega (t)}$.

• The convention here is the body-fixed one: $\mathrm{\Omega }=R^{\mathsf{T}}{R}^{\mathrm{\prime }}$ and $R^{\mathrm{\prime }}=R\mathrm{\Omega }$. The spatial angular velocity of the same path is the vector attached to $R^{\mathrm{\prime }}{R}^{\mathsf{T}}$, and the two differ by the rotation $R$; consumers must state which convention they use. For $C^2$ paths this $\omega$ is the same vector that appears in the rotating-frame acceleration identity of Angular velocity of a moving frame and the rotating-frame acceleration identity, whose clause 2 records it under a twice-differentiability hypothesis; the present lemma covers the $C^1$ case and proves only what is used here.

• The lemma is a statement about curves of matrices. No physical object, frame or time scale is introduced; an interpretation of $\omega$ as the angular velocity of a rigid body belongs to the physical items that adopt a rigid-body model.