= Antisymmetric first moment of a localized solenoidal field
{title2=$\int x_iJ_j\,dV=-\int x_jJ_i\,dV$}
For a <solenoidal vector field> $J$ with <compact support> and suitable integrability, apply the <divergence theorem> to $x_ix_jJ$. It gives zero boundary term and hence the displayed antisymmetry. Equivalently, for any constant <vector> $C$, the <vector triple product> identity yields $\int(C\cdot r)J\,dV=\tfrac12C\times\int(J\times r)\,dV$. A piecewise smooth field must have no normal jump at its support boundary; otherwise its <weak divergence> contains a surface contribution and the argument fails.
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