Given velocity vector {\vec {u}}=(u,v,w) the pathlines are defined as curves {\vec {r}}_s(t) = \big( \, x(s), \, y(s), \, z(s) \, \big) parametrized by parameter s:
| \begin{cases} \displaystyle {\frac {d{\vec {x}}_{P}}{dt}}(t)={\vec {u}}_{P}({\vec {x}}_{P}(t),t) \\[1.2ex] {\vec {x}}_{P}(t_{0})={\vec {x}}_{P0} \end{cases} |