Elementary Proofs Of The Gauss-Bonnet Theorem And Other Integral Formulas In $\Bbb R^3$

TL;DR

This paper provides straightforward, elementary proofs of the Gauss-Bonnet theorem, Poincaré-Hopf theorem, and other integral formulas for surfaces in three-dimensional space, avoiding the use of differential forms.

Contribution

It introduces simple, elementary proofs of key geometric theorems in $R^3$ that do not rely on differential forms, making the results more accessible.

Findings

Elementary proofs of Gauss-Bonnet and Poincaré-Hopf theorems

Proofs do not use fundamental or differential forms

Results applicable to compact surfaces with boundary in $R^3$

Abstract

For a compact differentiable surface with boundary embedded in $\Bbb R^3$, we give simple proofs of the Gauss-Bonnet theorem, Poincar\'{e}-Hopf theorem, and several other integral formulas. We complete all of the proofs without using fundamental or differential forms.