Final answer:
Buoyancy is not explicitly included in the fundamental equation of hydrostatics because it is the net effect of the pressure differences within the fluid. The local force of pressure gradient conceptually includes buoyancy since it accounts for the weight of the fluid element and the resulting pressure exerted by the fluid above it.
Step-by-step explanation:
The fundamental equation of hydrostatics, which can be expressed as →∇P=ρ→g, is derived from Newton's second law and does not explicitly include buoyancy because buoyancy is the net effect of pressure differences within the fluid. The equation P = ρgh tells us that pressure increases with depth in a static fluid, and this pressure difference contributes to the buoyant force. When considering a small fluid element within the body of the fluid, the forces due to gravity and the pressure gradient (the local force of pressure or →∇P) act on it. As these are local forces, they account for the weight of the fluid element and the resulting pressure exerted by the fluid above it, implicitly including the effect of buoyancy without directly naming it.
Buoyancy arises because the pressure at the bottom of a submerged object is greater than the pressure at the top, leading to a net upward force. This is consistent with Archimedes' principle, which states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. The fluid's own gravitational force on itself is considered when calculating the pressure difference, since this difference is what leads to buoyancy. Thus, in the equation →∇P=ρ→g, while buoyancy isn't directly mentioned, it is represented by the pressure gradient within the fluid.