I am studying PDEs and in some place I found an integral integrated by the surface measure on M=($C^k$ - Hypersurface of $R^n$).
1)Is there any reference to see how are defined this Measure on M?
2)What is the weakest assumption for the subset M of $R^n$ in order is defined the surface measure on M? ($C^k, k=?$)
3) When M is compact; is it different from the surface integral defined in Spivak Culculus on manifolds?
4) When one consider M as Riemannian manifold with the induced metric then this surface measure is equal to the Riemannian measure (GriGor'yan - Heat kernels and Analysis on Manifolds)?
Thanks in advance!!