Topics on Vector Calculus and Vector Analysis in $\mathbb {R}^n$

Abstract

In this chapter we address the main definitions and results for vector analysis in $\mathbb {R}^n$. This part of the text comprises topics such as differential forms in surfaces in $\mathbb {R}^n$, including the volume form for a surface in $\mathbb {R}^n$ and the Green, Gauss, and Stokes theorems in both standard calculus and abstract versions. Indeed we develop an abstract version of the Stokes theorem and recover the classical Divergence and Stokes theorems from such a general approach. We finish the chapter with an introduction to Riemannian geometry.