Weak divergence ($⟨\nabla \cdot J,\phi ⟩=-\int J\cdot \nabla \phi dV$)

The divergence defined through weak derivatives: pair a locally integrable vector field $J$ with a smooth compactly supported test function $\phi$ 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.