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Concept of Fluid Density

Fluid density is the mass per unit volume of a fluid, typically expressed in kilograms per cubic meter (kg/m³) or slugs per cubic foot (slugs/ft³).
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The statement of the theorem

Let Ω⊂R3\Omega \subset \mathbb{R}^3 be a continuous spatial domain occupied by the fluid, and let m(Ω)m(\Omega) be the total mass contained within Ω\Omega. The fluid density ρ(x⃗,t)\rho(\vec{x}, t) is defined as the material density function, which is a scalar field ρ:Ω×[t1,t2]→R+\rho: \Omega \times [t_1, t_2] \to \mathbb{R}^+. Formally, for any point x⃗∈Ω\vec{x} \in \Omega, the density ρ(x⃗,t)\rho(\vec{x}, t) is the limit of the ratio of mass mVm_{V} contained in a small volume VV surrounding x⃗\vec{x} to the volume VV itself, as VV approaches the zero volume limit: \nρ(x⃗,t)=lim⁡V→0mVV\rho(\vec{x}, t) = \lim_{V \to 0} \frac{m_{V}}{V} \nAlternatively, in the context of continuum mechanics, the mass mm contained within an arbitrary volume V⊂ΩV \subset \Omega is given by the volume integral of the density function: \nm=∭Vρ(x⃗,t) dVm = \iiint_{V} \rho(\vec{x}, t) \, dV \nFurthermore, the density is related to the fluid's equation of state, P=P(ρ,T)P = P(\rho, T), where PP is pressure, ρ\rho is density, and TT is temperature.