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Definition of Fluid Pressure

Fluid pressure is the force exerted per unit area by a fluid at rest, measured in Pascals (Pa) or pounds per square inch (psi).
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The statement of the theorem

In the context of a continuous fluid medium Ω⊂R3\Omega \subset \mathbb{R}^3 at rest (i.e., v⃗=0⃗\vec{v} = \vec{0}), the pressure PP is defined via the Cauchy stress tensor σ\boldsymbol{\sigma}. For an incompressible, static fluid, the stress tensor is isotropic and purely normal, σ=−PI\boldsymbol{\sigma} = -P \mathbf{I}, where I\mathbf{I} is the identity tensor. The pressure PP is mathematically defined as the scalar field representing the normal stress component acting on a surface element dAdA due to the fluid's internal energy and external body forces. Specifically, considering the equilibrium of a differential volume element dVdV, the Cauchy momentum equation reduces to: \n\n∇⋅σ+ρg⃗=0⃗\nabla \cdot \boldsymbol{\sigma} + \rho \vec{g} = \vec{0}\n\nSubstituting the hydrostatic stress assumption σ=−PI\boldsymbol{\sigma} = -P \mathbf{I} into the equilibrium equation yields: \n\n∇(−PI)+ρg⃗=0⃗\nabla (-P \mathbf{I}) + \rho \vec{g} = \vec{0}\n\nThis simplifies to the fundamental differential equation for pressure: \n\n∇P=−ρg⃗\nabla P = -\rho \vec{g}\n\nIntegrating this gradient yields the hydrostatic pressure equation, which defines PP at any point (x,y,z)(x, y, z) relative to a reference point (x0,y0,z0)(x_0, y_0, z_0): \n\nP(x,y,z)−P0=−ρg⃗⋅(r⃗−r⃗0)P(x, y, z) - P_0 = -\rho \vec{g} \cdot (\vec{r} - \vec{r}_0) \n\nWhere ρ\rho is the constant fluid density, g⃗\vec{g} is the acceleration due to gravity, and r⃗\vec{r} and r⃗0\vec{r}_0 are position vectors.