Differences

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--- misc4 [2018/08/16 16:28] – potthast
+++ misc4 [2023/03/28 09:14] (current) – external edit 127.0.0.1
@@ Line 3: / Line 3: @@
 $$
 \frac{dx'}{dt}(\rho,t)\Big|_{t=0} = 0   \hspace{3cm} (3.4)
+$$
+$$
+\frac{dx'}{dt}(\rho,t_0) = 0   \hspace{3cm} (3.4)
 $$
@@ Line 8: / Line 11: @@
 \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}