**A.1 Looking for the demonstration of Eq. (8)**

$$\frac{d}{dt}P = \frac{\partial}{\partial t}P + \frac{d}{d\infty}P\frac{\partial}{\partial t}\delta\mathbf{x} + \frac{d}{dy}P\frac{\partial}{\partial t}\delta y + \frac{d}{dz}P\frac{\partial}{\partial t}\delta\mathbf{z} \tag{A1}$$

$$\frac{d}{dt}P = \frac{\partial}{\partial t}P + \frac{d}{dx}Pv\_x + \frac{d}{dy}Pv\_y + \frac{d}{dz}Pv\_x \tag{A2}$$

$$\frac{d}{dt}P = \frac{\partial}{\partial t}P + \frac{d}{d\mathbf{x}}P v\_{\mathbf{x}} + \frac{d}{d\mathbf{y}}P v\_{\mathbf{y}} + \frac{d}{d\mathbf{z}}P v\_{\mathbf{z}} \tag{A3}$$

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$$\frac{d}{dt}P = \frac{\partial}{\partial t}P + \nabla P.v\tag{A4}$$

Since

$$\frac{d}{dt} = \frac{\partial}{\partial t} + v.\nabla \tag{A5}$$
