Write down the equations that are solved to give the path of a fluid particle in a flow with Eulerian fluid velocity $u(r,t)=(u,v,w).$

Also write down the equations that are solved to give the streamlines of the same flow.

Given the Eulerian velocity field

$3t(2xz,yt,z)$

write down and solve the streamline equations to determine the general form of the streamline at time $t($in the form $x=x(z,t),y=y(z,t).$ Deduce that at time $t=3$ the streamline through the point $(e,2,2)$ is given by

$x=e^{2z-3},y=\frac{z^3}{4}$