Differential Forms vs Geometric Algebra: The quest for the best geometric language

TL;DR

This paper compares differential forms and geometric algebra, two geometric formalisms in physics, by translating identities and applying them to electrodynamics and General Relativity to evaluate their relative strengths.

Contribution

It provides a side-by-side comparison and translation between differential forms and geometric algebra, clarifying their similarities and differences in physical applications.

Findings

Both formalisms can describe electrodynamics effectively

Geometric algebra offers a unified interpretation of geometric operations

Differential forms excel in expressing coordinate-free identities

Abstract

Differential forms is a highly geometric formalism for physics used from field theories to General Relativity (GR) which has been a great upgrade over vector calculus with the advantages of being coordinate-free and carrying a high degree of geometrical content. In recent years, Geometric Algebra appeared claiming to be a unifying language for physics and mathematics with a high level of geometrical content. Its strength is based on the unification of the inner and outer product into a single geometric operation, and its easy interpretation. Given their similarities, in this article we compare both formalisms side-by-side to narrow the gap between them in literature. We present a direct translation including differential identities, integration theorems and various algebraic identities. As an illustrative example, we present the case of classical electrodynamics in both formalism and finish with their description of GR.