= Complete boundary-jet condition for formal adjoints
{title2=$\int(Pu)v=\int u(P^*v)$}
For an order-$m$ operator with smooth coefficients on a bounded piecewise smooth domain, <integration by parts> has no boundary remainder when at each boundary point either the entire <boundary jet> of $u$ through order $m-1$ vanishes or the entire such jet of $v$ vanishes. Every boundary summand pairs <derivatives> of the two functions of order at most $m-1$. It is insufficient to require only that, for each same multi-index $\beta$, one of $\partial^\beta u$ and $\partial^\beta v$ vanish: for $P=\partial_x^2$, $u=x(a-x)\chi(y)$ and $v=1$ on a rectangle, all those same-index products vanish on the boundary but $\int(Pu)v=-2a\int\chi\ne0$.
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