Since fluids are almost always in motion, there is not all that much to fluid statics. Yay!
Really, the only bits are the hydrostatic equation (which is used as the basis of any fluid statics problem), pressure variation with depth, and forces on submerged surfaces – though these can be tricky.
The Hydrostatic Equation
The derivation of this equation is simple.
Take a fluid particle, where the area at the top and bottom is A and the height is δz
Taking the pressure at the bottom as P, we can work out the pressure at the top as P + the difference in pressure:
Using F = PA, calculate the force on the bottom and top:
The gravitational force of the fluid particle (the weight) is given as:
Now, we can balance the forces on the fluid particle:
This is the Hydrostatic Equation, and integrating both sides gives the equation for change in pressure with change in height that you are probably familiar with:
Note that you are probably familiar with ΔP = ρgh. This is the same, h is just defined as (z₁-z₂), hence the negative sign disappears. This form is sometimes known