Given the flow velocity field: $u(x,y,z,t)=2x(1+t),v(x,y,z,t)=4y,$ and $w(x,y,z,t)=0.$

a$)$ Find the element's path line and velocity vector as function of time $t$ only for an element that

passed through the point $x=1,y=1,z=1$ at $t=1.$ Construct a relationship $x_e=f_1(y_e)$ for the element's

path line.

b$)$ Find the equation $x=f_2(y)$ which describes the streamline that passes through the point $x=1,y=1,$

$z=1$ at $t=1.$

c$)$ Find the equation $x=f_3(y)$ for the streak line at time $t=1$ constructed by all fluid elements that

have passed through the point $x=1,y=1,z=1.$

d$)$ Plot all the above lines in a diagram $x$ vs$.y$ and compare between the lines.