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 {r}}_s(t)}{dt}}={\vec {u}}_s({\vec {r}}_s,t) \\[1.2ex] {\vec {r}}_s(t_{0})={\vec {r}}_{s0} \end{cases} |