Notes

For the sake of plotting and simple mathematical manipulations, I will occasionally use Maple.
You are not required to learn or use Maple; you can use anything you want to make plots
(including doing them by hand). For those who do want to use Maple, more information and
a tutorial are available here

Here is the Maple file illustrating the solution to  u_x+u_y+cu=0 with c=1, u(x,0)=exp(-x²/2).

Here are just the graphs and animations