From Leibniz's Theorem to the Reynolds Transport Theorem

The Taylor series expansion of \( F(\mathbf{x}, t) \) around \( t \) is given by:

\[ F(\mathbf{x}, t+\Delta t) = F(\mathbf{x}, t) + \frac{F'(\mathbf{x},t)}{1!} \Delta t + \frac{F''(\mathbf{x},t)}{2!} (\Delta t)^2 + \frac{F'''(\mathbf{x},t)}{3!} (\Delta t)^3 + \dots \]

For sufficiently small \( \Delta t \), keep only the first-order term:

\[ F(\mathbf{x}, t+\Delta t) \approx F(\mathbf{x}, t) + \frac{F'(\mathbf{x},t)}{1!} \Delta t. \]

This means that for small time increments, the change in \( F \) is linearly proportional to \( \Delta t \).

The first term inside the \(\{\}\)-braces can be expanded to four terms by defining the volume increment \( \Delta V \equiv V^*(t + \Delta t) - V^*(t) \) and applying a Taylor expansion to the integrand function:

\[ F(x, t + \Delta t) \approx F(x, t) + \Delta t \left( \frac{\partial F}{\partial t} \right) \]
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