From Foucault's Pendulum to the Gauss--Bonnet Theorem

TL;DR

This paper offers an accessible, self-contained proof of the Gauss-Bonnet theorem for 2D surfaces in R^3, using only classical vector calculus, aimed at advanced undergraduates and non-expert graduate students.

Contribution

It provides a novel, self-contained proof of the Gauss-Bonnet theorem utilizing only classical vector calculus, making the theorem more accessible to students.

Findings

Proof is self-contained and accessible

Uses only classical vector calculus techniques

Serves as educational illustration

Abstract

We present a self-contained proof of the Gauss-Bonnet theorem for two-dimensional surfaces embedded in $R^3$ using just classical vector calculus. The exposition should be accessible to advanced undergraduate and non-expert graduate students. It may be viewed as an illustration and exercise in multivariate calculus and a motivation to go deeper into the fields of geometry and topology.