The Exterior Derivative - A direct approach

TL;DR

This paper introduces a straightforward, determinant-based calculus approach to the exterior derivative, aiming to simplify its understanding in introductory courses compared to more abstract methods.

Contribution

It presents a direct, elementary method for defining the exterior derivative, contrasting with traditional abstract characterizations.

Findings

Simplifies the understanding of the exterior derivative for beginners.

Provides a determinant-based calculus lemma as a foundational tool.

Advocates for a direct approach in first courses on differential geometry.

Abstract

In this note we provide a direct approach to the most basic operator in this theory namely the exterior derivative. The crucial ingredient is a calculus lemma based on determinants. We maintain the view that in a first course at least this direct approach is preferable to the more abstract one based on characterization of the exterior derivative in terms of its properties.