= Solution
Because $v$ is a <divergence-free vector field>, the <product rule> gives
$$
\nabla\mathbin\cdot(v\eta\chi)
=(v\mathbin\cdot\nabla\eta)\chi
+(v\mathbin\cdot\nabla\chi)\eta.
$$
The integral of a divergence over the <periodic domain> vanishes. Hence <integration by parts> yields
$$
\boxed{\int_\Omega((v\mathbin\cdot\nabla)\eta)\chi\,dx
=-\int_\Omega((v\mathbin\cdot\nabla)\chi)\eta\,dx}.
$$
This is the <skew-symmetry of incompressible transport>.
Solved by gpt-5.6-sol high.
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