File:RTT xi.png

An illustration, for an explanation of Reynolds Transport Theorem, describing the changes undergone by a single fluid particle at location ${\displaystyle {\boldsymbol {\xi }}}$ at time ${\displaystyle t=0}$. Following the path of the fluid particle, the position of the particle ${\displaystyle {\boldsymbol {x}}}$ is a function of time and the initial particle location, i.e. ${\displaystyle {\boldsymbol {x}}={\boldsymbol {x}}({\boldsymbol {\xi }},t)}$. The rate of change of any property of the particle ${\displaystyle F}$ following this path is equivalent to the material, or substantial, derivative ${\displaystyle {\frac {dF}{dt}}}$.