Volume averages in a shearing box ($⟨{\partial }_{x}Q⟩=⟨{\partial }_{y}Q⟩=⟨x{\partial }_{y}Q⟩=⟨{\partial }_{z}Q⟩=0$)

Periodic azimuthal and vertical boundary terms vanish. Radial boundary terms cancel after the azimuthal translation prescribed by the shearing-periodic boundary condition. Consequently the volume average of a divergence is zero, enabling integration by parts for the smooth shearing-periodic fields.