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I'm struggling to understand one step in the derivation of the Reynold's Transport Theorem. I get as far as:

\begin{equation}

\frac{d}{dt} \int_{V(t)} f(\mathbf x, t) \, dV = \frac {d} {dt} \int_{V_0} f(\mathbf X,t) J(\mathbf X,t) \, dV = \int_{V_0} J \frac {\partial f} {\partial t} + f \frac {\partial J} {\partial t} \, dV

\end{equation}

From here, I want to write the RHS as:

\begin{equation}

\int_{V_0} \frac {\partial f} {\partial t} J + f \, ({\nabla \cdot} {\mathbf v}) J \,\, dV = \int_{V(t)} \frac {\partial f} {\partial t} + f \, ({\nabla \cdot} {\mathbf v}) \,\, dV

\end{equation}

However, all of my sources inexplicably replace the \begin{equation} \frac {\partial f} {\partial t} \end{equation} with a \begin{equation} \frac {Df} {Dt} \end{equation} Could anyone explain their reasoning or my error?

Thanks!

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# Reynolds Transport Theorem derivation question

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