▁from ▁the ▁coordinate ▁system ▁with ▁coordinate ▁variables ▁ ▁to ▁the ▁coordinate ▁system ▁with ▁coordinates ▁ ▁as : ▁ ▁Similarly ▁we ▁can ▁represent ▁ ▁as ▁a ▁function ▁of ▁ ▁as ▁follows : ▁▁ ▁for ▁▁ ▁similarly ▁we ▁can ▁write ▁the ▁free ▁equations ▁more ▁compact ly ▁as ▁▁ ▁for ▁▁ ▁These ▁three ▁equations ▁together ▁are ▁also ▁called ▁a ▁coordinate ▁transformation ▁from ▁ ▁to ▁. ▁Let ▁us ▁denote ▁this ▁transformation ▁by ▁. ▁We ▁will ▁represent ▁the ▁transformation ▁from ▁the ▁coordinate ▁system ▁with ▁coordinate ▁variables ▁ ▁to ▁the ▁coordinate ▁system ▁with ▁coordinates ▁ ▁as : ▁ ▁If ▁the ▁transformation ▁ ▁is ▁bi jective ▁then ▁we ▁call ▁the ▁image ▁of ▁the ▁transformation , nam ely ▁, ▁a ▁set ▁of ▁adm iss ible ▁coordinates ▁for ▁. ▁If ▁ ▁is ▁linear ▁the ▁coordinate ▁system ▁ ▁will ▁be ▁called ▁an ▁aff ine ▁coordinate ▁system ▁, other wise ▁ ▁is ▁called ▁a ▁cur vil ine ar ▁coordinate ▁system ▁ ▁The ▁Jacob ian ▁ ▁As ▁we ▁now ▁see ▁that ▁the ▁Co ordinates ▁▁ ▁and ▁ ▁are ▁functions ▁of ▁each ▁other , ▁we ▁can ▁take ▁the ▁derivative ▁of ▁the ▁coordinate ▁variable ▁ ▁with ▁respect ▁to ▁the ▁coordinate ▁variable ▁▁ ▁consider ▁▁ ▁for ▁, ▁these ▁derivatives ▁can ▁be ▁arranged ▁in ▁a ▁matrix , ▁say ▁, in ▁which ▁ ▁is ▁the ▁element ▁in ▁the ▁row ▁and ▁column ▁ ▁The ▁result ant ▁matrix ▁is ▁called ▁the ▁Jacob ian ▁matrix . ▁ ▁V ectors ▁in ▁cur vil ine ar ▁coordinates ▁Let ▁( b 1 , ▁b 2 , ▁b 3 ) ▁be ▁an ▁arbitrary ▁basis ▁for ▁three -