State and prove Archimedes principle.

Explanation

It states that when a body is partially or wholly immersed in a fluid, it immersed in a fluid, it experiences an upward thrust equal to the weight of the fluid displaced by it and its upthrust acts through the centre of gravity of the liquid displaced.
Upthrust or buoyant force = weight of liquid displaced.
Proof : It can be proved mathematically using Newton's Laws and Divergence Theorem. Consider a body of volume v having closed surface s submerged in liquid of density D.
Suppose the pressure at the surface of the fluid is zero then the pressure at a point 'P' at height y below the surface will be Dgy. Now, the net upward force exerted by the fluid on the body is the sum of vertical component of infinitesimal forces exerted on the body by the fluid over the surface.
This surface intergral becomes volume intergral of function 'Dg' (divergence of 'Dgy') by using of liqudi displaced) as the net upward force exerted on the body by the fluid.
Pressure function in the fluid,
P(x,y,z) `= rho g(H-y)`.
`vec(dF)` = Force exerted on the body by the fluid at the infitesimal surface area `vec(dA)`.
Let `vec(dF)_(y)` be the y-component of `vec(dF)` and `vec(P)`
`= rho g (H - y) hat(j)`
So, `vec(dF)_(y) = -(vec(dF) . vec(P)) hat(j)`
Magnitude of net buoyant force
`= oint ||vec(dF)_(y) || = oint_(A) - (vec(dP).vec(P))`
`= oint_(A)(-vec(P)).vec(dA)`
`= oint_(v) nabla .(-vec(P))d tau = oint_(v)nabla .(-rho g(H-y) hat(j)) d tau`
`= oint_(v) (-rho g(y - H)hat(j))d tau = oint_(v) rho g d tau`
`= rho g V` = weight of displaced fluid.