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

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.