# Exercises: Chapter 5, Section 5

1. Generalize the divergence theorem to the case of an -manifold with boundary in .

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 2. Applying the generalized divergence theorem to the set and , find the volume of in terms of the -dimensional volume of . (This volume is if is even and if is odd.)

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 .

3. Define on by and let be a compact three-dimensional manifold-with-boundary with . The vector field may be thought of as the downward pressure of a fluid of density in . Since a fluid exerts equal pressures in all directions, we define the buoyaant force on , due to the fluid, as . Prove the following theorem.

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.)