Pascal's law isotropic pressure

First feel the pressure gathered at one point. Each arrow is the pressure pushing on that point, and when you drag the direction the highlighted direction swings around while every arrow keeps the same length. The length is the size of the pressure, and it does not change however you turn the direction. Why? A fluid at rest cannot push harder in only one direction. If one direction were stronger, the fluid would have flowed that way. Sitting still without flowing means every direction is exactly in balance. What still water shows is a push that is the same from every side.

Prove the balance you just saw with one small wedge. Cut a very small triangle out of the fluid and pressure pushes perpendicular on each of the three faces, p_x on the vertical face, p_y on the bottom face, and p_n on the slanted face. Drag the angle to tilt the wedge and the areas of the faces each change, while the three pressures stay equal. The secret is balance of forces. The force on the slanted face is pressure times area, and the horizontal part of that force locks exactly with the force on the vertical face, and the vertical part with the force on the bottom. Geometry grows the area and the angle shaves off just as much, so in the end p_x = p_y = p_n. That is the root of Pascal's principle. After area and angle cancel each other, only the pressure remains.

So what exactly is pressure? It is the force pushing perpendicular on a face, divided by the area of that face, that is p = FA. It has the same shape as stress, but fluid pressure has two special points. First it always pushes perpendicular to the face, and second it always pushes and never pulls. Drag A and the force F stays the same, while a wider face lowers the pressure and turns it blue, and a narrower face raises it and turns it red. The same force crowded onto a narrow face sends the pressure up. A thumbtack tip that sinks in easily, and a wide ski that does not sink in snow, are both this division. A wide face scatters the force and a narrow face gathers it.

Now tie the earlier pieces into one. Take one small face through a point and turn that face, and whichever angle you turn it to, the pressure p that the face receives stays exactly the same. A value that does not change when you change the direction of the face means pressure has no direction. Force is a vector that has a direction, and pressure is a scalar that has none. So you write pressure as one number, not an arrow, and only when you decide which face it acts on do you multiply that number by the area and attach the normal direction of the face to get a force. Pressure itself is one state of that point. A face has to join the one number before a push appears.

This one fact hides all around. Choose a balloon and the air inside pushes every point of the skin equally and perpendicular, so the balloon goes round. A submersible has deep water pushing every part of the round shell equally and perpendicular, so a round section resists pressure best. Submerge a flat box and it is the same. Even if a wall is flat, at each point of that wall the pressure acts perpendicular to the face. Whatever the shape, a fluid pushes perpendicular on every face it touches, and at the same point it pushes with the same size in every direction. That is the ground that pressure with depth and buoyancy stand on. A round shell receives this push the most evenly.