▁E uclidean ▁distance ▁from ▁a ▁nearby ▁point ▁ ▁to ▁the ▁tang ent ▁plane ▁at ▁( x , ▁y ), ▁i . e . ▁the ▁length ▁of ▁the ▁per pend icular ▁dropped ▁from ▁the ▁nearby ▁point ▁to ▁the ▁tang ent ▁plane , ▁has ▁the ▁form ▁ ▁plus ▁third ▁and ▁higher ▁order ▁corre ctions . ▁The ▁above ▁expression , ▁a ▁symmetric ▁bil ine ar ▁form ▁at ▁each ▁point , ▁is ▁the ▁second ▁fundamental ▁form . ▁It ▁is ▁described ▁by ▁a ▁ 2   ×   2 ▁symmetric ▁matrix ▁ ▁which ▁depends ▁smooth ly ▁on ▁ ▁and ▁. ▁The ▁Gaussian ▁curv ature ▁can ▁be ▁calculated ▁as ▁the ▁ratio ▁of ▁the ▁determin ants ▁of ▁the ▁second ▁and ▁first ▁fundamental ▁forms : ▁ ▁Rem ark ably ▁G auss ▁proved ▁that ▁it ▁is ▁an ▁intr insic ▁invariant ▁( see ▁his ▁The or ema ▁E greg ium ▁below ). ▁ ▁One ▁of ▁the ▁other ▁extr insic ▁numerical ▁invari ants ▁of ▁a ▁surface ▁is ▁the ▁mean ▁curv ature ▁ ▁defined ▁as ▁the ▁sum ▁of ▁the ▁principal ▁curv atures . ▁It ▁is ▁given ▁by ▁the ▁formula ▁ ▁The ▁coefficients ▁of ▁the ▁first ▁and ▁second ▁fundamental ▁forms ▁satisfy ▁certain ▁compatibility ▁conditions ▁known ▁as ▁the ▁G auss - C od az zi ▁equations ; ▁they ▁involve ▁the ▁Christ off el ▁symbols ▁ ▁associated ▁with ▁the ▁first ▁fundamental ▁form : ▁ ▁These ▁equations ▁can ▁also ▁be ▁succ inct ly ▁expressed ▁and ▁derived ▁in ▁the ▁language ▁of ▁connection ▁forms ▁due ▁to ▁É lie ▁Cart an . ▁Pierre ▁Bon net ▁proved ▁that ▁two ▁quadratic ▁forms ▁satisfying ▁the