Consider the flow with velocity vector v = xi. Show that the individual particles have the position vectors r(t) = c₁e^ti + c₂j + c₃k with constant c₁, c₂, c₃. Show that the particles that at t = 0 are in the cube of Prob. 11 at t = 1 occupy the volume e.

Data from Prob. 11

Show that the flow with velocity vector v = yi is incompressible. Show that the particles that at time t = 0 are in the cube whose faces are portions of the planes x = 0, x = 1, y = 0, y = 1, z = 0, z = 1 occupy at t = 1 the volume 1.