Chapter 1 - Differential Calculus of Vector Fields and Differential Forms

Abstract

Chapter 1 presents the basic differential and integral relationships of the field theory. In this chapter, I also introduce the important concepts of work and flux of the field. Using these concepts, the different vector formulations of the Gauss's and Stokes's theorems are introduced and analyzed. Based on the Gauss's theorem, we develop a set of fundamental integral relations, which are known as Green's formulas. Another powerful mathematical approach to study the physical fields is based on the concept of differential forms. I provide the definitions and canonical representations of differential forms in three-dimensional Euclidean space. The chapter concludes by establishing a close relationship between the exterior differential operation of the calculus of the forms and the gradient, curl, and divergence operations of the conventional vector calculus.