Transformation Groups, Exterior Differentiation and Integration
In this chapter, the one-parameter transformation groups and Lie derivative are defined. Furthermore, two fundamental topics of differential geometry are presented in introductory form: exterior derivative and integration of r-forms. The exterior derivative extends to r-forms the elementary definitions of gradient of a function, curl, and divergence of a vector field as well as the meaning of exact and closed 1-forms. The integration of r-forms allows one to extend the definitions of surface and volume integrals as well as the theorems of Gauss and Stokes.