Pressure in fluids at a height/depth How can we prove that pressure in a system will be same in a particular fluid for the same height? Eg. in a hydraulic lift, and something even more complicated than that, can we always say that pressure at same height will be same. Imagine a random container which bends like a sine wave. So will the pressure be the same at 2 crests having same height?
Also, if I say that Pressure due to a liquid in a straightforward cylinder is Dgh, isn't that the pressure it applies downward because of its weight? Can we say that a depth 'h', even the pressure that the liquid applies to the sides of the container will be Dgh? Why will it be same for all directions at a particular height? 

 A: In the first paragraph, you're basically asking for a derivation of Bernoulli's equation:  https://en.wikipedia.org/wiki/Bernoulli%27s_principle#Derivations_of_Bernoulli_equation
Yes, no matter how complex the open container, if the system is static (not moving), the height of the liquid will be the same everywhere.  If one end is closed so that the "air" above the surface is not at atmospheric pressure, then the liquid height at that end can be different.  This is how a mercury barometer works.
Pressure is a scalar, not a vector.  It is direction-free.  It's the wall of the container that translates the pressure into a vector force.  $\vec{F} = P \vec{A}$ for a small area $\vec{A}$.  The direction is perpendicular to the surface and outward.
A: Great question. You have very valid queries, when you wonder how a downward force can exert pressure along the sides.
Spirko gave a great answer, but I'd like to share my 2 cents.
See this squeezy ball:

Even though you are pressing down, the squeezy ball reacts by expanding along the sides. This is a consequence of the objects tendency to maintain more-or-less a constant volume.
In the case of fluids it's a bit different, although even they tend to maintain a constant volume. Considering a bucketfull of water, it has many molecules of water in it. Each molecule feels the gravitational force, that attempts to pull it downward. As an analogy, consider each water molecule to be a person in this crowd:

Suppose every person suddenly decides to move forward (act of gravity). They would keep on moving on and on unless there's a wall in front of them. This is the case in free flowing water that falls from a height, like a waterfall:

However if there was a wall in front of the crowd, they can't move forward freely. However the people in the back don't know that there's a wall, and they keep shoving the people in front of them, in a hey why aren't y'all moving forward? kind of way. But of course, they'll just keep nudging around and no one moves anywhere. This is what happens in a bucketfull of still water. The water molecules nudge each other around, but as a whole, the crowd of water isn't really moving anywhere.
Back to analogy world, the people do feel nudged and shoved around. What would you do if you were shoved and nudged around? You would try to get out of the way. This matters more to the people in the front, since they can't move forward. So they tend to move towards the sides. But whoops, there are walls on the sides too (walls of the bucket). So basically, these poor people are banging on the walls, being pushed around by the crowd of people who all want to go forward. This is exactly how pressure arises. In our analogy world, we could translate pressure to the nudginess a person feels, and the force exerted on the walls of container as a result of the helpless person being shoved towards the walls.
Now, suppose someone opens a hidden door on the side. Whoosh, goes the angry mob out of the door in a hurry to get away from the madness they were subjected to. Water molecules are no different. If you punch a hole along the sides of a bucket, the water streams out, just like the crowd in analogy world.
In some ways, water molecules behave like people (albeit without personality and uniqueness), although that wasn't the point that I was about to convey here.
I hope I addressed the OP in a way he wanted it.
