Distributional divergence ($⟨\mathrm{div}z,\eta ⟩=-\int z\cdot \nabla \eta$)

For a locally integrable vector field, this formula defines its divergence as a distribution. For a piecewise smooth field, a jump of its normal component across an interface adds a surface measure to the smooth divergences. A continuous normal component eliminates that measure, as follows by integration by parts on the two sides.