Volume average ($⟨f{⟩}_V=V^{-1}{\int }_{V}f{d}^{3}x$)

A volume average is the normalized integral average of a field over a three-dimensional region of finite positive volume. For a fixed region, differentiating in time commutes with this average under the usual regularity conditions. The divergence theorem converts the average of a divergence into its net boundary flux divided by volume.