The normal fluid density is defined through the moment of inertia of a cylinder rotating about its axis, in the limit of infinitesimal angular velocity. The angular momentum may be calculated from the one particle Green function, and this leads to the relation between ρn and the Green function. We then consider an ensemble in which the fluid is constrained to irrotational flow. For statistical states which are “locally gauge invariant” we show that ρn, as defined previously, is identical to the full density. This demonstrates the connection between superfluidity and the breaking of local gauge symmetry. Finally, we apply the result to the gas of non-interacting quasi-particles and the fully interacting Bose fluid in the limit of very low temperatures. In both cases we reproduce the Landau expression for ρn in terms of the quasiparticle occupation numbers, and derive the well-known result for real superfluid helium that ρn ~ T4 as T→0.