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Given velocity vector 

LaTeX Math Inline
body{\vec {u}}=(u_x,u_y,u_z)
 the pathlines is a bunch of curves 
LaTeX Math Inline
body{\vec {r}}_s = \big( \, x_s, \, y_s, \, z_s \, \big) \in \mathbb{R}^3
, parametrized by real number the stem points
LaTeX Math Inline
body{\vec {r}}_{s0}
 at zero-time
LaTeX Math Inline
bodyst_0
, and solving the following equation:

LaTeX Math Block
alignmentleft
\begin{cases}


\displaystyle 


{\frac {d{\vec {xr}}_{P}s(t)}{dt}}(t)={\vec {u}}_{P}s({\vec {xr}}_{P}(t)s,t)
\\[1.2ex]

{\vec {xr}}_{P}s(t_{0})={\vec {xr}}_{P0s0}

\end{cases}