Prove the Bernoulli theorem

The sum of pressure energy , kinetic energy  and potential energy of liquid remains constant 

Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.

Derivation : 
Let the velocity, pressure and area of a fluid column at a point A be v₁, p and A₂ and at another point B be v₂, p₂, p₂ and A₂ .

Let the volume that is bounded by A and B be moved to M and N .

let AM = L₁ and BN = L₂ .

Now if we can compress the fluid then we have,

A₁L₁ = A₂L₂

Net work done per volume = p₂ - p₁

Kinetic energy per volume = 1/2 ρ v²

Kinetic energy gained per volume = 1/2 ρ (v₂² - v₁²) 

Potential energy gained per volume = ρ g(h₂ - h₁) 

Now , 

p₁ - p₂ = 1/2 ρ (v₂² - v₁²)  + ρ g(h₂ - h₁) 

p₁ - p₂ = 1/2 ρ v₂² - 1/2 ρ v₁² + ρ g h₂ - ρ g h₁

p₁ +  1/2 ρ v₁²  + ρ g h₁ = p₂ + 1/2 ρ v₂² + ρ g h₂

p + 1/2 ρ v + ρ g h = constant 

Applications : 

The top of the airplane wing is a little curved and the bottom is completely flat. But air travels across both parts of the wings simultaneously in the sky.

The entire pitch of the baseball is working on the principle of Bernoulli’s theorem. The stitches of the ball can be seen forming a curve which makes it necessary for the pitcher to grip the ball’s seams.