= Weak divergence
{title2=$\langle\nabla\cdot J,\varphi\rangle=-\int J\cdot\nabla\varphi\,dV$}
The <divergence> defined through <weak derivatives>: pair a locally integrable <vector field> $J$ with a smooth compactly supported <test function> $\varphi$ using the displayed integration-by-parts identity. A piecewise smooth field can have a surface contribution where its normal component jumps. Thus zero classical <divergence> on each side does not imply zero weak divergence across the interface.
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