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will be given only by expression (2), when we don't apply it to the mentioned sphere, but to an arbitrary formed larger space.

Of course ${\bar {\phi }}=\phi$ everywhere, as soon as $\phi$ doesn't show rapid changes.

Furthermore we easily find

${\frac {\partial {\bar {\phi }}}{\partial t}}={\frac {\overline {\partial \phi }}{\partial t}}{,}\ {\frac {\partial {\bar {\phi }}}{\partial x}}={\frac {\overline {\partial \phi }}{\partial x}}{,}{\text{ u. s. w.}}$

m. By the average of a vector ${\mathfrak {A}}$ we understand a vector — it may be called ${\bar {\mathfrak {A}}}$ —, whose components are the averages of ${\mathfrak {A}}_{x},{\mathfrak {A}}_{y},{\mathfrak {A}}_{z}$. Consequently we have

${\dot {\bar {\mathfrak {A}}}}={\bar {\dot {\mathfrak {A}}}},\ Div\ {\bar {\mathfrak {A}}}={\overline {Div}}\ {\bar {\mathfrak {A}}},\ Rot\ {\bar {\mathfrak {A}}}={\overline {Rot}}\ {\bar {\mathfrak {A}}}.$