Ok, so I'm trying to get the reading done early...

To answer the question at the end of the first lecture notes, I'm pretty sure it is possible - you just need u_t to be equal to -(u \cdot \nabla) u.

The second lecture notes were interesting, it was nice to see the Euler and Nanvier-Stokes equations derived again. I guess these aren't quite the same Euler equations that we had before, since we're assuming incompressibility whereas before there'd been one equation with derivatives of the density. I guess it's a reasonable and useful assumption for a lot of cases.