I was thinking about why the buoyant force on an object should depend solely on it's volume and not shape. It seems loosely like the divergence theorem in that an integral over the surface is determined by the volume. There is a big difference though; in the divergence theorem we integrate scalars (flux/divergence) but to find the buoyant force we must integrate a vector. By making the vector-analogous arguments behind the divergence theorem, I am led to the following conclusion. Suppose you have a scalar field P, and a volume V with surface S(V), then I believe: S(V)∫∫ P dA = V∫∫∫ ∇P dV (where dA is the outward pointing area-element) In the example of buoyancy, the scalar field P is the pressure P = C-ρgz with C being constant, ρ being the fluid density, g being the gravitational field strength, and +z being vertically upwards. Then ∇P = -ρgez. The net force due to fluid pressure is F = ∫∫ P (-dA) = -∫∫∫ ∇S dV = -∫∫∫(-ρgez) dV = ρgez∫∫∫ dV = (ρV)gez; Which is Archimedes principle. (The -dA is because force comes from integrating P over the inward pointing normal.) I am wondering what is the name of this theorem I have stated? Is it somehow just a restatement of the divergence theorem? (If so; how to get between the two?) I cannot find anything about it.