Note, the density term is in the pressure gradient force because this is the force per unit mass (so we get rid of ill defined volumes in the atmosphere).

In the class derivation/discussion of this, I had a mild brain fart. I don't know why I kept writing the pressure gradient in velocity coordinates instead of the horizontal spatial coordinate (n or x in the above). The above is correct.

The following simple picture might help better visualize this. The horizontal velocity is largest in zone A, then zone B, and finally zone C as the pressure gradient is higherst in zone A and lowest in zone B, along the horizontal coordinate x: