==== Maths Notation ====

$$
\frac{dx'}{dt}(\rho,t)\Big|_{t=0} = 0   \hspace{3cm} (3.4)
$$
$$
\frac{dx'}{dt}(\rho,t_0) = 0   \hspace{3cm} (3.4)
$$

$$
\frac{dy'}{dt}(\rho,t)\Big|_{t=0} = x(\rho,0)   \hspace{3cm} (3.5)
$$
$$
\frac{dy'}{dt}(\rho,t_0) = x(\rho,t_0) = x_0   \hspace{3cm} (3.5)
$$

\begin{eqnarray}
y'(\rho,t) & = &  y'(\rho,t) - y'(\rho,t_0) \\
& = &  \int_{t_0}^t \frac{dy'(\rho,s)}{ds} \; ds \\
& = &  \frac{dy'(\rho,t_0)}{dt} \cdot \delta t + O(\delta t^2)\\
& = & x(\rho,t_0)\cdot \delta t + O(\delta t^2)
\end{eqnarray}