Curl of Linear Flows

Although we sometimes treat a vector field as a static object, you can also think of a vector field as being the right-hand side of a differential equation. From this perspective, the vector field F=(F1,F2,F3) corresponds to the differential equation dx/dt=F1, dy/dt=F2, and dz/dt=F3. Streamlines of the vector field correspond to trajectories of the differential equation.

Each of the following movies shows the evolution of streamlines for a linear velocity field. Assume the third component of the velocity field is identically zero, that is, F3=0. Thus is makes sense to think about these flows as being two-dimensional (convince yourself that this is true!).

For each movie below, answer the questions in Question #1.

• Movie #1: A flow in the middle of a deep channel
• Movie #2: A flow near the shore of a channel. (The "shore: is at y=0.)
• Movie #3: A flow approximating a linear vortex (or whirlpool)

Question #1

1. Based on the trajectories that you see in the movie, write down a formula for a vector field that has similar trajectories. For example, in the first movie, the trajectories move at constant speed in the x direction and do not move at all in the y direction. Trajectories for the differential equation dx/dt=1 and dy/dt=0 have this property, so we can model the vector field by F=(1,0,0). (The last component was chosen to be zero by the assumption above.) Hint: pay close attention to the range of the variables when formulating your equations.
2. Compute the curl for your model of the vector field. Which direction does the curl point?
3. For each point (x,y,z), write down a function S so that S(x,y,z) is the speed of the trajectory at (x,y,z). Hint: recall that speed is the magnitude of velocity. For example, if we model the velocity field in the first movie by F(x,y,z)=(1,0,0), then S(x,y,z)=1 since all trajectories move at unit speed.

Question #2