How can we prove the work done by gas expansion in any deformable container?

[raopeng]

In my textbook W=∫p.dV is only proved for a syringe with a piston. This is quite easily done but the book never explains how it extrapolates to the general situation for a gas expanding in any deformable container. It seems the point is to prove dV= S.h where S is the surface area of a given container and h is a small displacement alone the direction of the normal to the surface. I tried in this line of thought but things soon get really mathematical and confusing. How can prove this formula in any deformable containers? Thanks in advance!

 

[raopeng]

Well the mathematical idea is that since for a (n-1) sub-manifold a R^n-1 can be introduced, the orthogonal complement to the hyperplane R^n-1 becomes after a suitable diffeomorphism the normal to the surface in the space, Hence it is possible to write dV=S.h for any n-volume. Then dW=F.h=p.S.h=p.dV...