Pascal's law and pressure in fluid at a depth

We were introduced fluid mechanics in class today I am thoroughly confused on some basic points-

• What is the proof of Pascal's law? And what does it exactly mean? I know that it states that the pressure is transmitted equally to every point in fluid. Let's take a point in the left beaker at depth h from the piston. So does it have a pressure of P(atm) + P + rhogh? (P(atm)= atmospheric pressure). If so, then that pressure must be same throughout the horizontal line at h. So if the piston in right beaker is at height h' from that horizontal line, does that imply P(atm) + P + rhogh= P(atm) + P+ rhogh'? h=h'? So what does it really mean to be transmitted to all points? Does it mean that pressure at every point will increase by P than original value?

• Why is pressure even rhogh at depth h? My understanding is that pressure in fluid is really just the normal force the layer applies on upper layers as in the table applies on a book and the book applies the pressure back on table. So if it is a uniform cylinder then it makes sense that the weight above the layer at h must be rhoghA and the pressure rhog*h. What about when shape changes? Why is pressure same at all points at a depth? Then the weight above layer isn't equal to what mentioned above. It just doesnt make sense no matter how many answers I read here or on other sites.

• Proof for archimedes principle is simple enough if it is a cylinder or cuboid by taking pressure difference at top and bottom. What about irregular shapes?

• When a container accelerates on an incline with acceleration other than g*sin(theta) where theta is the angle of inclined plane, why is the water surface slanted? Can you explain with an FBD?

Please don't close this question. I know it's a bit too basic but I have tried to understand these answers for hours and I can't find a satisfying explanation elsewhere.

When working with a hydraulic system, it is often assumed that the pressure is high enough that you can ignore the differences in height. Notice in your equation that the atmospheric pressure drops out. (That may not be true in a different situation.) Then you are left with a pressure difference of ρg(h-h'). Archimedes principle is based on the fact that if you remove an object which is subject to a buoyant force, then the volume of fluid that had been displaced (but now returns), must now be supported by the same buoyant force. That is independent of the shape of the volume displaced.

• But what's the proof of archimedes principle? And in the Pascal's law diagram does the pressure increase by P at every point? Apr 8, 2021 at 15:28
• What would consider more proof than simple logic? In a fluid, the pressure is the same at every point that is at the same height. Apr 8, 2021 at 19:01