Conceptual understanding for pressure inside incompressible fluid I'm was thinking about incompressible fluid and how pressure effects it and I came across this idea that I'm not sure if it's correct. I would like some correction along with my current understanding. Thank you in advanced.
From what I know so far (which may be wrong) is when you have a object placed in water, the bouyancy is equal to the weight of water deplaced by that object. If we then have a cup with water (let's say height h) when we place an object inside the water that doesn't fully drown, it will make the water level from the cup a bit higher (say h+a). I feel like if this is possible the reverse should also be possible. If our object is a rectangle, I think by putting a cover that has a hole that's size of rectangle's side area and pushing on water will make the water level go down (until it reaches original height h) and make the rectangle float up.
Now, is this possible? If it is, can I get more explanation about how the pressure outside of water can affect this? Is it possible that the atmospheric is so high that human can walk on water?
 A: Since water is incompressible, it's volume is constant. This means, if your cover has the area $A_c$ and the area of the bottom side of the cuboid is $A_o$, then pushing the cover down by $a$ means decreasing the volume beneath it by $aA_c$. In order to conserve volume, the cuboid has to rise by $aA_c/A_o$, so what you describe will indeed happen. The nice thing about this is, that one does not even have to think about force or pressure; this is a purely geometric problem.
Now about the outside pressure. The thing one needs to keep in mind is that in most situations, it will impose a force both on the cover and the cuboid. As a consequence, it won't be possible to walk on water, unless most mass of the atmosphere is concentrated below your shoulders (in which case the pressure would probably crush you), because it pushes you downward just (almost) as much as the water.
Remark: Actually, at very high pressures, water can of course be further compressed, however those pressures are normally not encountered in everyday life.
