The pressure exerted by a static fluid is same in all directions.

Consider a small cubic element of fluid as shown.

Fluid element is in equilibrium so, forces acting on each face are equal in magnitude.

Areas of each face are equal therefore, pressure of fluid on each lateral face is same

In the limit as the cube element reduces to a point, pressure at top and bottom surface become equal.
Hence, pressure exerted by fluid is same in all directions - the pressure is isotropic.

The pressure is same at all points on the same horizontal level in a static fluid (or moving with uniform velocity or with vertical acceleration)

Consider a cylindrical volume element of fluid within the fluid contained in a vessel.

If `Delta S` be the cross-section of cylinder, then force `F_1` on face `A` is `P_1(Delta S)` `P_1` and force `F_2` on face `B` is `P_2 (Delta S)` `P_1` and `P_2` are respective pressures at face `A` and `B` respectively.

In equilibrium,

`F_1 = F_2 => P_1 (Delta S) = P_2 (Delta S)`

or `P_1 = P_2`

On any horizontal plane in static fluid, pressure is same at all the points.

Consider a fluid contained in a beaker. Let force exerted by fluid on bottom of beaker is along direction RP.

From Newton's third law, force exerted by beaker on fluid is along PR. Resolving this force along two rectangular components PX and PY.

Since fluid cannot resist tangential force, the fluid at P should begin to flow along PX parallel to surface.

But fluid is at rest, so there cannot be any component along PX.

The force exerted by a fluid on any surface is normal to the surface and is called thrust of fluid.

The thrust per unit area of the surface in contact with fluid is called Pressure.

Consider a fluid contained in a container exposed to atmosphere. Let a small volume element at depth (h) of area (A), thickness (dh) in the fluid.

`P_0 =` Atmospheric Pressure = pressure acting on free surface of fluid

F = Total force acting downward on top face of volume element

`F=PxxA`

`F' =` Total force acting upward on bottom face of volume element

`F' = (P + dP ) xx A`

`dW =` weight of this element in equilibrium,
`F + dW = F'`

`dW = F' - F = (P + dP) A- PA , dW = (dP)A`

`(dP)A =dW=(dhA) rho g`

`dP = dh rho g`

`int_0^P dP = int_0^h rho dh`
`P = P_0 + h rho g`

Pressure is same on all points at the same depth.

Total pressure at depth `(h)` from free surface is greater than atmospheric pressure.

Density `(rho)` of the fluid remain constant throughout its volume. It is good assumption for liquids.Pressure at depth `(h)` can be obtained by the relation

`dP = dh rho g`

`int_(P_0)^(P) dP = int_0^h rho g dh = rho g int_0^h dh `

`P = P_0 +h rho g`

where , `rho =` density of fluid

`P_0 =` Atmospheric Pressure

Absolute pressure is the total pressure at a point.

Gauge pressure is the pressure relative to the local atmospheric pressure.

`P_( a b s o l u t e) = P_(a t m) + P_(g a u g e)`

Gauge pressure may be positive, zero or negative.