PlanetPhysics/Reynolds Transport Theorem

Let $$F(\mathbf{r},t)$$ 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 $${\cal V}$$ of the material is obtained through the volume integral. $$ \int_{\cal V} F(\mathbf{r},t) \;dV $$

If this property is being transported by the action of the flow of the material with a velocity $$\mathbf{u}(\mathbf{r},t)$$, 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 $${\cal S}$$ (bounding $${\cal V}$$) to and from the surrounding region. $$ \frac{d}{d t} \int_{\cal V} F(\mathbf{r},t) \;dV = \int_{\cal V} \frac{\partial F}{\partial t} \;dV + \int_{\cal S} F\mathbf{u} \cdot \mathbf{n} \;dS $$ Here, $$\mathbf{n}$$ is a unit vector indicating the normal direction of the surface (oriented to point out of the volume).