Differential Forms and Applications

Abstract

This project deals mainly with Differential Forms on smooth Riemannian manifolds and their applications through the properties of their classical Differential and Integral Operators. The calculus of Differential Forms provides a simple and flexible alternative to vector calculus. It is not dependent on any coordinate system, simplifies or condenses variational principles, offers a more comprehensive means of evaluating multivariate integrals, and is crucial in the analysis of the variation of differentiable functions on smooth manifolds. Differential Forms have numerous applications within (and beyond) Differential Geometry and Mathematical Physics. Needless to mention, Differential Forms constitute the ingredients (test functions) of the Theory of k-current which is analogous to Distribution Theory, and so they offer diverse potential tools for research.