▁with ▁origin ▁ ▁and ▁, ▁ ▁the ▁rad ii ▁at ▁polar ▁angles ▁ 0 ▁and ▁: ▁ ▁where ▁the ▁second ▁equality ▁follows ▁from ▁the ▁G auss – Jac obi ▁equation ▁and ▁the ▁fourth ▁from ▁G auss ' ▁derivative ▁formula ▁in ▁the ▁orthogonal ▁coordinates ▁. ▁ ▁G auss ' ▁formula ▁shows ▁that ▁the ▁curv ature ▁at ▁a ▁point ▁can ▁be ▁calculated ▁as ▁the ▁limit ▁of ▁angle ▁excess ▁ ▁over ▁area ▁for ▁success ively ▁smaller ▁ge odes ic ▁tri angles ▁near ▁the ▁point . ▁Qual it atively ▁a ▁surface ▁is ▁posit ively ▁or ▁neg atively ▁cur ved ▁according ▁to ▁the ▁sign ▁of ▁the ▁angle ▁excess ▁for ▁arbitr arily ▁small ▁ge odes ic ▁tri angles . ▁ ▁G auss – B on net ▁theorem ▁ ▁Since ▁every ▁compact ▁orient ed ▁ 2 - man if old ▁ ▁can ▁be ▁tri ang ulated ▁by ▁small ▁ge odes ic ▁tri angles , ▁it ▁follows ▁that ▁ ▁where ▁ ▁denotes ▁the ▁E uler ▁characteristic ▁of ▁the ▁surface . ▁ ▁In ▁fact ▁if ▁there ▁are ▁ ▁faces , ▁ ▁edges ▁and ▁ ▁vertices , ▁then ▁ ▁and ▁the ▁left ▁hand ▁side ▁equals ▁. ▁ ▁This ▁is ▁the ▁celebrated ▁G auss – B on net ▁theorem : ▁it ▁shows ▁that ▁the ▁integral ▁of ▁the ▁Gaussian ▁curv ature ▁is ▁a ▁topological ▁invariant ▁of ▁the ▁manifold , ▁namely ▁the ▁E uler ▁characteristic . ▁This ▁theorem ▁can ▁be ▁interpreted ▁in ▁many ▁ways ; ▁perhaps ▁one ▁of ▁the ▁most ▁far - re aching ▁has ▁been ▁as ▁the ▁index ▁theorem ▁for ▁an ▁elli ptic ▁differential ▁operator