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Fluid pressure is the magnitude of force per unit area exerted by a fluid. For incompressible fluids like water, pressure can be expressed using the fluid's specific weight or density. The relationship is expressed mathematically as p = ρgz, where ρ is fluid density, g is gravitational acceleration, and z is depth from the surface.

Pressure increases linearly with depth because the weight of fluid above a point accumulates as you move deeper. At the same depth, all points experience identical pressure. However, at shallower depths, less fluid weight exists above, resulting in lower pressure. This linear relationship holds for incompressible fluids but not gases, whose density changes with temperature and pressure.

Pascal's law states that pressure intensity at any point in an incompressible static fluid is the same in all directions. This principle applies to fluids at rest and is fundamental to hydraulic system design. Engineers use this law to ensure systems can withstand forces created by fluid pressure without damage or failure.

Consider a wedge-shaped infinitesimal fluid element at rest. Forces on each side equal pressure multiplied by area. Since the liquid is stationary, horizontal and vertical force components must sum to zero. Using trigonometric relations and simplifying these equilibrium equations yields px = ps and py = ps, proving pressure is identical in all directions.

Pressure at a given depth depends directly on fluid density and gravitational acceleration. The equation p = ρgz shows that higher density fluids generate greater pressure at the same depth. This relationship is valid for incompressible liquids but does not apply to gases, where density varies significantly with both temperature and pressure changes.

Engineers use Pascal's law and pressure-depth relationships to design hydraulic systems that safely handle fluid forces. Understanding that pressure acts equally in all directions and increases linearly with depth allows engineers to calculate required material strength and component specifications. This ensures systems withstand operational pressures without failure.

The incompressibility assumption allows engineers to use the linear pressure-depth equation p = ρgz reliably. For incompressible fluids like most liquids, density remains constant regardless of pressure or depth changes. This simplifies calculations significantly. However, this assumption fails for gases, whose density changes considerably with temperature and pressure variations.