= Reciprocity of the clamped-beam Green function
{title2=$G(x,\xi)=G(\xi,x)$}
The <clamped-beam Green function> is symmetric. Its explicit formula depends only on $x_<=\min(x,\xi)$ and $x_>=\max(x,\xi)$. More generally, two <integrations by parts> give $\int uv''''dx=\int u''v''dx=\int u''''v\,dx$ when $u=u'=v=v'=0$ at both endpoints. Applying this identity to two point-load responses shows that the displacement at one point caused by a unit force at a second equals the displacement at the second caused by a unit force at the first.
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