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

For a compact differentiable surface with boundary embedded in ${\mathbb{R}}^3$, 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.