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Relationship between Pressure and Depth

Pressure is directly proportional to the fluid density and the depth, assuming constant gravity.
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The statement of the theorem

The relationship is derived from the fundamental principle of fluid statics, which states that in a fluid at equilibrium, the pressure gradient must balance the body forces. Assuming a fluid of density ρ\rho and constant gravitational acceleration g⃗=−gk^\vec{g} = -g\hat{k} (where k^\hat{k} is the unit vector in the vertical direction, and zz is the vertical coordinate increasing upwards), the governing differential equation is:\∇P=−ρg⃗\nabla P = -\rho \vec{g}. Since ρ\rho and gg are assumed constant, and the pressure PP is a function of position r⃗=(x,y,z)\vec{r} = (x, y, z), we have:\∂P∂x=0\frac{\partial P}{\partial x} = 0, \∂P∂y=0\frac{\partial P}{\partial y} = 0, and \∂P∂z=−ρg\frac{\partial P}{\partial z} = -\rho g. Integrating the partial derivative with respect to zz yields the hydrostatic pressure profile:\dPdz=−ρg\frac{dP}{dz} = -\rho g. Integrating this differential equation from the surface (z=0z=0) to an arbitrary depth zz gives the pressure P(z)P(z) at depth zz: \∫PatmP(z)dP=−ρg∫0zdz\int_{P_{atm}}^{P(z)} dP = -\rho g \int_{0}^{z} dz. \\Therefore, the pressure P(z)P(z) at depth zz is given by the equation:\P(z)=Patm+ρgh\mathbf{P(z) = P_{atm} + \rho g h},where, where h = -zisthedepthmeasuredpositivelydownwards,and is the depth measured positively downwards, and P_{atm}istheatmosphericpressureatthesurface( is the atmospheric pressure at the surface (z=0).If). If hisdefinedasthedepth,therelationshipis:$P(h)=Patm+ρgh is defined as the depth, the relationship is:\$\mathbf{P(h) = P_{atm} + \rho g h}.
Source: Wikipedia