From Vector Analysis to Differential Forms

TL;DR

This paper simplifies the derivation of exterior differentiation using differential forms, building on previous work that characterized divergence and curl through the divergence and Stokes theorems at the infinitesimal level.

Contribution

It provides a simplified, differential forms-based derivation of exterior differentiation, connecting classical vector analysis with modern geometric methods.

Findings

Exterior differentiation derived naturally from differential forms.

Unified treatment of divergence and curl via differential forms.

Reinforces the geometric foundation of differential operators.

Abstract

In our previous paper [International Journal of Theoretical Physics, 41 (2002), 1165-1190] we have shown, following the tradition of synthetic differential geometry, that div and rot are uniquely determined, so long as we require that the divergence theorem and the Stokes theorem should hold on the infinitesimal level. In this paper we will simplify the discussion considerably in terms of differential forms, leading to the natural derivation of exterior differentiation in the usual form.