PlanetPhysics/Reynolds Transport Theorem

Let $F(𝐫,t)$ represent the amount of some physical property of a continuous material medium per unit [[../Volume/|volume]]. The total amount of this property present in a finite region $𝒱$ of the material is obtained through the volume integral.

$\underset{𝒱}{\int }F(𝐫,t)dV$

If this property is being transported by the action of the flow of the material with a [[../Velocity/|velocity]] $𝐮(𝐫,t)$, then Reynolds' transport theorem states that the rate of change of the total amount of $F$ within the material volume is equal to the volume integral of the instantaneous changes of $F$ occuring within the volume, plus the surface integral of the rate at which $F$ is being transported through the surface $𝒮$ (bounding $𝒱$) to and from the surrounding region.

$\frac{d}{dt}\underset{𝒱}{\int }F(𝐫,t)dV=\underset{𝒱}{\int }\frac{\partial F}{\partial t}dV+\underset{𝒮}{\int }F𝐮\cdot 𝐧dS$

Here, $𝐧$ is a [[../PureState/|unit vector]] indicating the normal direction of the surface (oriented to point out of the volume).