Consider the following two$-$dimensional velocity field:

vec$(v)=-$yhat$(i)+$xhat$(j)$

a$)$ Use the forward Euler method to draw a pathline $($which is also a streamline$)$ through the

point $(x,y)=(2,0).$ Show streamlines for time step sizes $t=0\mathrm{.}2$ and $0\mathrm{.}1.$ How quickly does

your answer improve as $t$ is lowered? Include your code and derivations in your solutions.

b$)$ Repeat the previous part using the predictor$-$corrector method. Discuss any differences. which

method is more accurate?

c$)$ Show that this velocity field satisfies conservation of mass for an incompressible flow and

determine the stream function.

d$)$ Plot the streamline through the same point as in parts a and b$,$ this time using the fact

that the streamline is a contour of the stream function. Give the $($implicit$)$ equation for this

streamline, and compare your plot to parts a and b$.$