Covector Fields and Cartan’s Exterior Differential – the Beauty of Differential Forms

Abstract

The calculus of differential forms was introduced by Élie Cartan (1869–1951) in 1899. It was Cartan’s goal to study Pfaff systems

$$\sum _{k=1}^n a_{jk}(x ^1, \ldots, x^n) dx^k=0, \qquad j= 1, \ldots, m$$

by using a symbolic method. It turns out that:

Cartan’s calculus is the proper language of generalizing the classical calculus due to Newton (1643–1727) and Leibniz (1646–1716) to real and complex functions with n variables.

The key idea is to combine the notion of the Leibniz differential df with the alternating product a∧b due to Grassmann (1809–1877). Cartan’s calculus has its roots in physics. It emerged in the study of point mechanics, elasticity, fluid mechanics, heat conduction, and electromagnetism. It turns out that Cartan’s differential calculus is the most important analytic tool in modern differential geometry and differential topology, and hence Cartan’s calculus plays a crucial role in modern physics (gauge theory, theory of general relativity, the Standard Model in particle physics). In particular, as we will show in Chap. 19, the language of differential forms shows that Maxwell’s theory of electromagnetism fits Einstein’s theory of special relativity, whereas the language of classical vector calculus conceals the relativistic invariance of the Maxwell equations.