Pascal's law

Pascal's law: the pressure at a point in a fluid at rest is the same in all directions, i.e. $p_x = p_y = p_z$.

Proof

Consider a small wedge-shaped fluid element at rest, of unit width perpendicular to the paper, with sides $dx$ (horizontal), $dy$ (vertical) and inclined face $ds$ at angle $\theta$ to the vertical.

          |\
          | \  ds  (pressure p_z
   p_x -> |  \       normal to it)
       dy |   \
          |    \
          |_____\
            dx
            ^ p_y

Pressures: $p_x$ on the vertical face, $p_y$ on the bottom face, $p_z$ on the inclined face. Geometry: $dy = ds\cos\theta$, $dx = ds\sin\theta$ (with $\theta$ the angle between the inclined face and the vertical).

Since the fluid is at rest, there are no shear forces; only pressure forces and weight act.

Horizontal equilibrium:

$$p_x (dy \cdot 1) - p_z (ds \cdot 1)\cos\theta = 0$$

Using $ds\cos\theta = dy$: $p_x\,dy = p_z\,dy \Rightarrow p_x = p_z$.

Vertical equilibrium:

$$p_y (dx\cdot 1) - p_z (ds\cdot 1)\sin\theta - \rho g \frac{dx\, dy}{2}\cdot 1 = 0$$

Using $ds\sin\theta = dx$:

$$p_y\,dx - p_z\,dx - \rho g \frac{dx\,dy}{2} = 0$$

As the element shrinks to a point ($dy \to 0$), the weight term (second-order small) vanishes, so $p_y = p_z$.

Therefore

$$p_x = p_y = p_z$$

Since $\theta$ was arbitrary, the pressure at a point is the same in every direction.

Applications

Hydraulic press, hydraulic jack and brakes, where a small force on a small piston gives a large force on a large piston ($F_2 = F_1 A_2/A_1$).