Given velocity vector {\vec {u}}=(u_x,u_y,u_z) the pathlines is a bunch of curves {\vec {r}}_s = \big( \, x_s, \, y_s, \, z_s \, \big) \in \mathbb{R}^3, parametrized by real number s, and solving the following equation:
| \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} |