Archimedes' principle can be deduced mathematically from Euler's equations "Euler's equations (fluids)") for a fluid at rest, which in turn can be deduced by generalizing Newton's laws to a continuous medium. In the same way, Archimedes' principle can be deduced from the Navier-Stokes equations for a fluid:

The condition that the incompressible fluid is at rest implies taking into the previous equation, which allows us to arrive at the fundamental relationship between fluid pressure, fluid density and acceleration of gravity:

From that relationship we can easily rewrite the forces on a submerged body in terms of the weight of the fluid displaced by the body. When a solid K is immersed in a fluid, at each point on its surface a force appears per unit area perpendicular to the surface at that point and proportional to the pressure of the fluid p at that point. If we call the vector normal to the surface of the body we can write the resultant of the forces simply using Stokes' divergence theorem:

where the last equality occurs only if the fluid is incompressible.

Suppose a body of volume immersed in a fluid of density, we can now choose small elements of area, such that they tend to be a point on the surface of the body.

A value pressure and a force associated with it act on each point (area element), such that.

All the forces that are surrounding the body due to pressure at the same level are canceled. leaving only forces in downward and upward directions.

Now if we take two points on the surface of the body that are connected through a vertical we have a respective force downward and another upward and therefore a respective resultant.

Where the part is a small volume element of the body, .

Therefore, it can be rewritten as:.

Now, the thrust becomes the net force.

Where the sum of all the small volume elements of the body, , turns out to be the total volume of the submerged body, i.e.,.

Therefore we arrive at:

That is, the thrust is proportional to the volume of the fluid displaced by the body, that is, proportional to the volume of the submerged body.

Knowing that , replacing we obtain:.

That is, the thrust is equal to the weight of the displaced fluid.

With this, Archimedes' principle is demonstrated.

For a right prism of base A and height H, immersed in a completely vertical position, the previous demonstration is really elementary. Due to the configuration of the prism within the fluid, the pressures on the lateral area only produce horizontal thrusts that, in addition, cancel each other and do not contribute to sustaining it. For the upper and lower faces, since all its points are submerged to the same depth, the pressure is constant and we can use the relationship force = pressure × area, and taking into account the result on the upper and lower face, we have:.

where is the pressure applied on the lower face of the body, is the pressure applied on the upper face and A is the projected area of ​​the body. Taking into account the general equation of hydrostatics, which states that the pressure in a fluid at rest increases proportionally with depth:

Introducing the volume of the body in the last term and multiplying by the density of the fluid ρ we see that the upward vertical force F is precisely the weight of the displaced fluid.

The thrust or force that the liquid exerts on a body, vertically and upwardly, when it is submerged, also turns out to be the difference between the weight that the body has suspended in the air and the "weight" that it has when it is introduced into a liquid. The latter is known as the "apparent" weight of the body, since its weight in the liquid "apparently" decreases; The force that the Earth exerts on the body remains constant, but the body, in turn, receives an upward force that decreases the vertical resultant.

where is the weight of the body in the air and is the weight of the body immersed in the liquid.

Archimedes' principle does not take into account the surface tension (capillarity) acting on the body.[10]​ Furthermore, it has been discovered that Archimedes' principle breaks down in complex fluids.[11]​ Complex fluids are mixtures in which two phases coexist: solid-liquid ("Suspension (chemistry)" suspensions or solutions of macromolecules such as polymers), solid-gas (granular), liquid-gas (foams) or liquid-liquid (emulsions).

There is an exception to Archimedes' principle known as the inferior (or lateral) case. This occurs when one side of the object is touching the bottom (or side) of the container in which it is submerged, and no liquid is seeping along that side. In this case, it has been proven that the net force differs from Archimedes' principle, because by not filtering liquid on that side, the symmetry of the pressure is broken.[12]​.

Archimedes' principle shows the buoyant force and displacement of a fluid. However, the concept of Archimedes' principle can be applied when considering why objects float. Proposition 5 of Archimedes' treatise On Floating Bodies states that.

In other words, for an object floating on a liquid surface (such as a ship) or floating submerged in a fluid (such as a submarine in water or a blimp in air) the weight of the displaced liquid is equal to the weight of the object. Therefore, only in the special case of flotation is the buoyant force acting on an object equal to its weight. Let's consider a 1 ton block of solid iron. Because iron is almost eight times denser than water, it only displaces 1/8 of a ton of water when submerged, which is not enough to keep it afloat. Suppose the same block of iron is transformed into a bowl. It still weighs 1 ton, but when submerged in water, it displaces a greater volume of water than when it was a block. The deeper the iron bowl is submerged, the more water it displaces and the greater the buoyant force acting on it. When the buoyant force is equal to 1 ton, it no longer sinks.

When a boat displaces a weight of water equal to its own weight, it floats. This is often called the "flotation principle": A floating object displaces a weight of fluid equal to its own weight. Every ship, submarine and airship must be designed to displace a weight of fluid at least equal to its own weight. The hull of a 10,000 ton ship must be wide, long and deep enough to displace 10,000 tons of water and still have some hull above the water to prevent it from sinking. It needs additional hull to combat the waves that would otherwise fill it and, by increasing its mass, cause it to submerge. The same goes for ships in the air: an airship weighing 100 tons needs to displace 100 tons of air. If it moves more, it rises; If it moves less, it falls. If the airship displaces exactly its weight, it floats at a constant altitude.

Although related, the principle of buoyancy and the concept that a submerged object displaces a volume of fluid equal to its own volume are not Archimedes' principle. Archimedes' principle, as already mentioned, equates the buoyant force to the weight of the displaced fluid.

A common point of confusion regarding Archimedes' principle is the meaning of displaced volume. The most common demonstrations involve measuring the rise in water level when an object floats on the surface to calculate the water displaced. This measurement method fails with a submerged floating object because the rise in water level is directly related to the volume of the object and not the mass (unless the effective density of the object is exactly equal to the density of the fluid).[14]​[15]​[16]​.

Archimedes' principle can be deduced mathematically from Euler's equations "Euler's equations (fluids)") for a fluid at rest, which in turn can be deduced by generalizing Newton's laws to a continuous medium. In the same way, Archimedes' principle can be deduced from the Navier-Stokes equations for a fluid:

The condition that the incompressible fluid is at rest implies taking into the previous equation, which allows us to arrive at the fundamental relationship between fluid pressure, fluid density and acceleration of gravity:

From that relationship we can easily rewrite the forces on a submerged body in terms of the weight of the fluid displaced by the body. When a solid K is immersed in a fluid, at each point on its surface a force appears per unit area perpendicular to the surface at that point and proportional to the pressure of the fluid p at that point. If we call the vector normal to the surface of the body we can write the resultant of the forces simply using Stokes' divergence theorem:

where the last equality occurs only if the fluid is incompressible.

Suppose a body of volume immersed in a fluid of density, we can now choose small elements of area, such that they tend to be a point on the surface of the body.

A value pressure and a force associated with it act on each point (area element), such that.

All the forces that are surrounding the body due to pressure at the same level are canceled. leaving only forces in downward and upward directions.

Now if we take two points on the surface of the body that are connected through a vertical we have a respective force downward and another upward and therefore a respective resultant.

Where the part is a small volume element of the body, .

Therefore, it can be rewritten as:.

Now, the thrust becomes the net force.

Where the sum of all the small volume elements of the body, , turns out to be the total volume of the submerged body, i.e.,.

Therefore we arrive at:

That is, the thrust is proportional to the volume of the fluid displaced by the body, that is, proportional to the volume of the submerged body.

Knowing that , replacing we obtain:.

That is, the thrust is equal to the weight of the displaced fluid.

With this, Archimedes' principle is demonstrated.

For a right prism of base A and height H, immersed in a completely vertical position, the previous demonstration is really elementary. Due to the configuration of the prism within the fluid, the pressures on the lateral area only produce horizontal thrusts that, in addition, cancel each other and do not contribute to sustaining it. For the upper and lower faces, since all its points are submerged to the same depth, the pressure is constant and we can use the relationship force = pressure × area, and taking into account the result on the upper and lower face, we have:.

where is the pressure applied on the lower face of the body, is the pressure applied on the upper face and A is the projected area of ​​the body. Taking into account the general equation of hydrostatics, which states that the pressure in a fluid at rest increases proportionally with depth:

Introducing the volume of the body in the last term and multiplying by the density of the fluid ρ we see that the upward vertical force F is precisely the weight of the displaced fluid.

The thrust or force that the liquid exerts on a body, vertically and upwardly, when it is submerged, also turns out to be the difference between the weight that the body has suspended in the air and the "weight" that it has when it is introduced into a liquid. The latter is known as the "apparent" weight of the body, since its weight in the liquid "apparently" decreases; The force that the Earth exerts on the body remains constant, but the body, in turn, receives an upward force that decreases the vertical resultant.

where is the weight of the body in the air and is the weight of the body immersed in the liquid.

Archimedes' principle does not take into account the surface tension (capillarity) acting on the body.[10]​ Furthermore, it has been discovered that Archimedes' principle breaks down in complex fluids.[11]​ Complex fluids are mixtures in which two phases coexist: solid-liquid ("Suspension (chemistry)" suspensions or solutions of macromolecules such as polymers), solid-gas (granular), liquid-gas (foams) or liquid-liquid (emulsions).

There is an exception to Archimedes' principle known as the inferior (or lateral) case. This occurs when one side of the object is touching the bottom (or side) of the container in which it is submerged, and no liquid is seeping along that side. In this case, it has been proven that the net force differs from Archimedes' principle, because by not filtering liquid on that side, the symmetry of the pressure is broken.[12]​.

Archimedes' principle shows the buoyant force and displacement of a fluid. However, the concept of Archimedes' principle can be applied when considering why objects float. Proposition 5 of Archimedes' treatise On Floating Bodies states that.

In other words, for an object floating on a liquid surface (such as a ship) or floating submerged in a fluid (such as a submarine in water or a blimp in air) the weight of the displaced liquid is equal to the weight of the object. Therefore, only in the special case of flotation is the buoyant force acting on an object equal to its weight. Let's consider a 1 ton block of solid iron. Because iron is almost eight times denser than water, it only displaces 1/8 of a ton of water when submerged, which is not enough to keep it afloat. Suppose the same block of iron is transformed into a bowl. It still weighs 1 ton, but when submerged in water, it displaces a greater volume of water than when it was a block. The deeper the iron bowl is submerged, the more water it displaces and the greater the buoyant force acting on it. When the buoyant force is equal to 1 ton, it no longer sinks.

When a boat displaces a weight of water equal to its own weight, it floats. This is often called the "flotation principle": A floating object displaces a weight of fluid equal to its own weight. Every ship, submarine and airship must be designed to displace a weight of fluid at least equal to its own weight. The hull of a 10,000 ton ship must be wide, long and deep enough to displace 10,000 tons of water and still have some hull above the water to prevent it from sinking. It needs additional hull to combat the waves that would otherwise fill it and, by increasing its mass, cause it to submerge. The same goes for ships in the air: an airship weighing 100 tons needs to displace 100 tons of air. If it moves more, it rises; If it moves less, it falls. If the airship displaces exactly its weight, it floats at a constant altitude.

Although related, the principle of buoyancy and the concept that a submerged object displaces a volume of fluid equal to its own volume are not Archimedes' principle. Archimedes' principle, as already mentioned, equates the buoyant force to the weight of the displaced fluid.

A common point of confusion regarding Archimedes' principle is the meaning of displaced volume. The most common demonstrations involve measuring the rise in water level when an object floats on the surface to calculate the water displaced. This measurement method fails with a submerged floating object because the rise in water level is directly related to the volume of the object and not the mass (unless the effective density of the object is exactly equal to the density of the fluid).[14]​[15]​[16]​.