Now, we know that pressure in a fluid depends only upon depth, and any increase in pressure at the surface must be transmitted to every point in the field. This is known as Pascal’s law as it was first recoginsed by the French Scientist Blaise Pascal.

According to this law, pressure exerted on any fluid at any place is transmitted equally in all directions and is always normal to the container. This law is also called the law of transmission of pressure.
Proof of Pascal’s law: We can prove Pascal’s law by using two principles:
(i) The force on any layer of a fluid at rest is normal to the layer, and
(ii) Newton’s 1st law of motion.
We consider a small element ABCDEF in the form of a right angled prism inside a vessel containing fluid at rest.
The effect of gravity is same for all points in the element. We consider that fluid exerts pressure Pa, Pb and Pc on the forces BEFC, ADFC and ADEB respectively of this element and corresponding normal forces on these faces are Fa, Fb and Fc.
Let Aa, Ab and Ac be the respective areas of the three faces. In right angled ∆ABC, let ∠ACB = θ
Now we consider different forces in equilibrium,
Horizontally, Fb sinθ = Fc ............(1)
Vertically Fb cosθ = Fa ..............(2)
Also, from the geometry of the figure, we get
Ab sinθ = Ac .........(3)
and Ab cosθ = Aa .........(4)
From equations (1), (2), (3) and (4)

This proves Pascal's law
1. Hydraulic Brakes: In hydraulic brakes, when a little force with the foot is applied on the pedal (see fig), the pressure so applied gets transmitted through brake oil to act on a larger area where pistons move the brake-shoes against the brake-drums. So on applying a little pressure on the pedal a large force is exerted on the wheel which produces deceleration.

2. Hydraulic Lift: One can see at motor service station that the cars and large trucks being raised to convenient heights so that mechanic can work under them. Here, a slight pressure is transmitted through the liquid to act on a large surface, thus, producing sufficient force to raise up the vehicle. If F1 is the force acting on the piston of area A1, then the pressure below that piston is \(P = \frac{F_1}{A_1}\)

According to the Pascal’s law, the pressure on any area element in the large piston will be same. If the cross-sectional area of this cylinder is A2, then the total force acting on the larger piston will be
F2 = P A2 or P = \(\frac{F_2}{A_2}\)
So, \(\frac{F_1}{A_1} = \frac{F_2}{A_2}\)
The gain in force will be,
\(\frac{F_2}{F_1} = \frac{A_2}{A_1}\)
This ratio is also called mechanical advantage.