Eureka!

A proof that the net force due to pressure on a (fully or partially) submerged object is equal to the weight of the water displaced.

The facts/approximations used were

• pressure is isotropic
• density is constant
• gravitational force is constant

And the method of proof was to apply the Divergence Theorem.

POSTSCRIPT

I was just wondering why pressure is isotropic when I found this post and  I realised that the scratchings above actually prove it!  Imagine a blob of water submerged in water: the above proof shows that the net force on it is zero, but only if pressure is isotropic!