▁ ▁and ▁ ▁are ▁analyt ic ; ▁soon ▁afterwards ▁É lie ▁Cart an ▁general ised ▁this ▁to ▁local ▁embed d ings ▁of ▁Riemann ian ▁- man if olds ▁in ▁ ▁where ▁. ▁To ▁prove ▁Jan et ' s ▁theorem ▁near ▁( 0 , 0 ), ▁the ▁C auch y – K ow ale v ski ▁theorem ▁is ▁used ▁twice ▁to ▁produce ▁analyt ic ▁ge odes ics ▁orthogonal ▁to ▁the ▁- axis ▁and ▁then ▁the ▁- axis ▁to ▁make ▁an ▁analyt ic ▁change ▁of ▁coordinate ▁so ▁that ▁ ▁and ▁. ▁An ▁is ometric ▁embedding ▁ ▁must ▁satisfy ▁. ▁D iffer enti ating ▁gives ▁the ▁three ▁additional ▁equations ▁. ▁with ▁ ▁and ▁ ▁pres cribed . ▁These ▁equations ▁can ▁be ▁solved ▁near ▁( 0 , 0 ) ▁using ▁the ▁C auch y – K ow ale v ski ▁theorem ▁and ▁yield ▁a ▁solution ▁of ▁the ▁original ▁embedding ▁equations . ▁ ▁Orth ogonal ▁coordinates ▁When ▁ ▁in ▁the ▁metric , ▁lines ▁parallel ▁to ▁the ▁- ▁and ▁- ax es ▁are ▁orthogonal ▁and ▁provide ▁orthogonal ▁coordinates . ▁If ▁, ▁then ▁the ▁Gaussian ▁curv ature ▁is ▁given ▁by ▁ ▁If ▁in ▁addition ▁, ▁so ▁that ▁, ▁then ▁the ▁angle ▁ ▁at ▁the ▁intersection ▁between ▁ge odes ic ▁ ▁and ▁the ▁line ▁ ▁= ▁constant ▁is ▁given ▁by ▁the ▁equation ▁ ▁The ▁derivative ▁of ▁ ▁is ▁given ▁by ▁a ▁classical ▁derivative ▁formula ▁of ▁G auss : ▁ ▁Ge odes ic ▁polar ▁coordinates ▁▁ ▁Once ▁a ▁metric ▁is ▁given ▁on ▁a ▁surface ▁and ▁a ▁base ▁point ▁is ▁fixed , ▁there