Proving Archimedes' principle from the fluid pressure gradient

Step 1 of 4: Pressure grows with depth

Intuition

Each water layer must 'carry' all the water above it — the deeper you go, the heavier the load, so pressure grows evenly with depth, like a stack of boxes.

p(h) = p_0 + \rho\,g\,h

Analysis

Balance a small fluid column of cross-section $S$ and height $dh$ : the upward pressure force $p\,S$ on its base must carry the downward pressure on its top plus the column's weight $\rho g S\,dh$ , giving $dp = \rho g\,dh$ ; integrating yields $p(h) = p_0 + \rho g h$ with $h$ measured from the free surface.