For \( \Delta t \to 0 \), approximate the integral: \( \int_{V^*(t+\Delta t)} F(\mathbf{x},t+\Delta t) dV \approx \int_{V^*(t)} F(\mathbf{x},t) dV + \int_{V^*(t)} \Delta t \frac{\partial F(\mathbf{x},t)}{\partial t} dV + \int_{\Delta V} F(\mathbf{x},t) dV + \int_{\Delta V} \Delta t \frac{\partial F(\mathbf{x},t)}{\partial t} dV\)
This simplifies to: \[ \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x},t) \, dV = \lim_{\Delta t \to 0} \frac{1}{\Delta t} \left\{ \int_{V^*(t)} \Delta t \frac{\partial F(\mathbf{x},t)}{\partial t} \, dV + \int_{\Delta V} F(\mathbf{x},t) \, dV \right\} \]
and this limit may be taken once the relationship between \( \Delta V\) and \( \Delta t\) is known.
Substituting into the time derivative equation: \( \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x},t) dV = \lim\limits_{\Delta t \to 0} \frac{1}{\Delta t} \left( \int_{V^*(t+\Delta t)} F(\mathbf{x},t+\Delta t) dV - \int_{V^*(t)} F(\mathbf{x},t) dV \right)\)
Rearranging:
\( \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x},t) dV = \lim\limits_{\Delta t \to 0} \frac{1}{\Delta t} \bigg( \cancel{\int_{V^*(t)} F(\mathbf{x},t) dV} + \int_{V^*(t)} \Delta t \frac{\partial F(\mathbf{x},t)}{\partial t} dV + \int_{\Delta V} F(\mathbf{x},t) dV + \int_{\Delta V} \Delta t \frac{\partial F(\mathbf{x},t)}{\partial t} dV - \cancel{\int_{V^*(t)} F(\mathbf{x},t) dV} \bigg)\)
Since \( \Delta t \) is a constant factor: \( \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x},t) dV = \lim\limits_{\Delta t \to 0} \left( \int_{V^*(t)} \frac{\partial F(\mathbf{x},t)}{\partial t} dV + \frac{1}{\Delta t} \int_{\Delta V} F(\mathbf{x},t) dV + \int_{\Delta V} \frac{\partial F(\mathbf{x},t)}{\partial t} dV \right)\)
Since \( \int_{\Delta V} \frac{\partial F(\mathbf{x},t)}{\partial t} dV \) is of order \( \mathcal{O}(\Delta t) \), it vanishes as \( \Delta t \to 0 \), leading to the final equation: \( \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x},t) dV = \int_{V^*(t)} \frac{\partial F(\mathbf{x},t)}{\partial t} dV + \lim\limits_{\Delta t \to 0} \frac{1}{\Delta t} \int_{\Delta V} F(\mathbf{x},t) dV\)
This simplifies to: \( \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x},t) dV = \int_{V^*(t)} \frac{\partial F(\mathbf{x},t)}{\partial t} dV + \int_{\Delta V} F(\mathbf{x},t) dV\)