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At fixed time write the azimuthal shift as $\delta =2A{L}_{x}t$ modulo $L_y$. Integrating a radial derivative over the box gives a boundary difference proportional toThe two integrals agree by translation invariance of an integral over a full azimuthal period. Hence $⟨{\partial }_{x}Q⟩=0$. The ordinary periodic boundary conditions make the $y$ and $z$ boundary differences zero as well. Finally,for every $x,z$. Thus the volume averages in a shearing box satisfyFor smooth shearing-periodic extensions, spatial derivatives and products inherit the same shearing-periodic boundary condition. Their divergence therefore has zero volume average, and integration by parts is available without a residual boundary flux. Time differentiation of an average over this fixed physical box commutes with integration, even though the radial identification shifts with time.