The generalization: Let be a compact -dimensional manifold-with-boundary and the unit outward normal on . Let be a differentiable vector fieldd on . Then
As in the proof of the divergence theorem, let . Then . By Problem 5-25, on , we have for . So,
By Stokes' Theorem, it follows that
One has and since the outward normal is in the radial direction. So . In particular, if , this says the surface area of is times the volume of .
Theorem (Archimedes). The buoyant force on is equal to the weight of the fluid displaced by .
The definition of buoyant force is off by a sign.
The divergence theorem gives . Now is the weight of the fluid displaced by . So the right hand side should be the buoyant force. So one has the result if we define the buoyant force to be . (This would make sense otherwise the buoyant force would be negative.)