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

2017/01/06 by Stoytchev, Orlin
#Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #History and Overview (math.HO)

paper · doi:10.48550/arxiv.1701.01666

Abstract

We present a self-contained proof of the Gauss-Bonnet theorem for two-dimensional surfaces embedded in R³ 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.

Related

• Gauss--Bonnet theorem for compact and orientable surfaces: a proof without using triangulations
• Elementary Proofs Of The Gauss-Bonnet Theorem And Other Integral Formulas In \Bbb R³
• Gaussian curvature in codimension > 1
• The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature
• Philosophical Reflections on Intrinsic Differential Geometry around the Gauss-Bonnet Theorem