Let
represent the amount of some physical property of a continuous material medium per unit volume. The total amount of this property present in a finite region
of the material is obtained through the volume integral.
If this property is being transported by the action of the flow of the material with a velocity
, then Reynolds' transport theorem states that the rate of change of the total amount of
within the material volume is equal to the volume integral of the instantaneous changes of
occuring within the volume, plus the surface integral of the rate at which
is being transported through the surface
(bounding
) to and from the surrounding region.
Here,
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