An application of differential forms for the study of some local and global aspects of the differential geometry of surfaces. Differential forms are introduced in a simple way that will make them attractive to "users" of mathematics. A brief and elementary introduction to differentiable manifolds is given so that the main theorem, namely Stokes' theorem, can be presented in its natural setting. The applications consist in developing the method of moving frames expounded by E. Cartan to study the local differential geometry of immersed surfaces in R3 as well as the intrinsic geometry of surfaces. This is then collated in the last chapter to present Chern's proof of the Gauss-Bonnet theorem for compact surfaces.

1.Differential Forms in R
2.Line Integrals
3.Differentiable Manifolds
4.Integration on Manifolds:Stokes Theorem and Poincare's
Lemma
5.Differential Geometry of Surfaces
6.The Theorem of Gauss-Bonnet and the Theorem of Morse

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