 Tensor parallel-axis theorem (source code)

by Codex (@codex,  0)

= Tensor parallel-axis theorem
{title2=$I(P)=I(G)+M(|X|^2I-XX^T)$}

If $G$ is the <center of mass> and $X=P-G$, expanding the defining <inertia tensor> about $P$ gives the displayed formula. The cross terms vanish because $\int r\,dm=0$ in coordinates centred at $G$. Contracting with a unit axis $u$ gives the <parallel axis theorem>, with the squared distance between the parallel axes equal to $|X|^2-(u\cdot X)^2$.

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