▁the ▁older ▁scientific ▁literature ▁in ▁mechan ics ▁and ▁physics ▁and ▁can ▁be ▁ind isp ens able ▁to ▁understanding ▁work ▁from ▁the ▁early ▁and ▁mid ▁ 1 9 0 0 s , ▁for ▁example ▁the ▁text ▁by ▁Green ▁and ▁Z erna . ▁ ▁Some ▁useful ▁relations ▁in ▁the ▁algebra ▁of ▁vectors ▁and ▁second - order ▁tens ors ▁in ▁cur vil ine ar ▁coordinates ▁are ▁given ▁in ▁this ▁section . ▁ ▁The ▁notation ▁and ▁contents ▁are ▁primarily ▁from ▁Og den , ▁Nag h di , ▁Sim mond s , ▁Green ▁and ▁Z erna , ▁ ▁Bas ar ▁and ▁We ich ert , ▁and ▁C iar let . ▁ ▁Co ordinate ▁transformations ▁ ▁Consider ▁two ▁coordinate ▁systems ▁with ▁coordinate ▁variables ▁ ▁and ▁, ▁ ▁which ▁we ▁shall ▁represent ▁in ▁short ▁as ▁just ▁ ▁and ▁ ▁respectively ▁and ▁always ▁assume ▁our ▁index ▁ ▁runs ▁from ▁ 1 ▁through ▁ 3 . ▁We ▁shall ▁assume ▁that ▁ ▁these ▁coordinates ▁systems ▁are ▁embedded ▁in ▁the ▁three - dimensional ▁e uclidean ▁space . ▁Co ordinates ▁▁ ▁and ▁ ▁may ▁be ▁used ▁to ▁explain ▁each ▁other , ▁because ▁as ▁we ▁move ▁along ▁the ▁coordinate ▁line ▁in ▁one ▁coordinate ▁system ▁we ▁can ▁use ▁the ▁other ▁to ▁describe ▁our ▁position . ▁In ▁this ▁way ▁Co ordinates ▁▁ ▁and ▁ ▁are ▁functions ▁of ▁each ▁other ▁▁ ▁for ▁▁ ▁which ▁can ▁be ▁written ▁as ▁▁▁ ▁for ▁▁ ▁These ▁three ▁equations ▁together ▁are ▁also ▁called ▁a ▁coordinate ▁transformation ▁from ▁ ▁to ▁ ▁. Let ▁us ▁denote ▁this ▁transformation ▁by ▁. ▁We ▁will ▁therefore ▁represent ▁the ▁transformation