微分形式与几何代数：寻找最佳几何语言
Differential Forms vs Geometric Algebra: The quest for the best geometric language
AI总结：
本文比较微分形式与几何代数两种几何形式体系，给出直接转换及恒等式，并以经典电动力学和广义相对论为例展示其应用。
摘要：
微分形式是物理学中一种高度几何化的形式体系，从场论到广义相对论（GR）均有应用。它是对矢量微积分的重大升级，具有无坐标和高度几何内容的优势。近年来，几何代数出现，声称是物理学和数学的统一语言，具有高度的几何内容。其优势在于将内积和外积统一为单一的几何运算，且易于解释。鉴于它们的相似性，本文对两种形式体系进行了并排比较，以缩小文献中它们之间的差距。我们给出了直接转换，包括微分恒等式、积分定理和各种代数恒等式。作为示例，我们展示了两种形式体系下的经典电动力学，并以它们对GR的描述作为结束。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
