▁explicit ▁expression ▁for ▁the ▁transformation ▁tensor ▁F ▁because ▁an ▁alternative ▁form ▁of ▁the ▁mapping ▁between ▁cur vil ine ar ▁and ▁Cart esian ▁bases ▁is ▁more ▁useful . ▁ ▁Assuming ▁a ▁sufficient ▁degree ▁of ▁smooth ness ▁in ▁the ▁mapping ▁( and ▁a ▁bit ▁of ▁ab use ▁of ▁notation ), ▁we ▁have ▁ ▁Similarly , ▁ ▁From ▁these ▁results ▁we ▁have ▁ ▁and ▁ ▁Vector ▁and ▁tensor ▁calculus ▁in ▁three - dimensional ▁cur vil ine ar ▁coordinates ▁ ▁Sim mond s , ▁in ▁his ▁book ▁on ▁tensor ▁analysis , ▁quotes ▁Albert ▁Ein stein ▁saying ▁▁ ▁The ▁magic ▁of ▁this ▁theory ▁will ▁hardly ▁fail ▁to ▁imp ose ▁itself ▁on ▁anybody ▁who ▁has ▁truly ▁understood ▁it ; ▁it ▁represents ▁a ▁genu ine ▁triumph ▁of ▁the ▁method ▁of ▁absolute ▁differential ▁calculus , ▁founded ▁by ▁G auss , ▁Riemann , ▁Ric ci , ▁and ▁Le vi - C iv ita . ▁Vector ▁and ▁tensor ▁calculus ▁in ▁general ▁cur vil ine ar ▁coordinates ▁is ▁used ▁in ▁tensor ▁analysis ▁on ▁four - dimensional ▁cur vil ine ar ▁manif olds ▁in ▁general ▁relativ ity , ▁in ▁the ▁mechan ics ▁of ▁cur ved ▁shell s , ▁in ▁exam ining ▁the ▁in var iance ▁properties ▁of ▁Max well ' s ▁equations ▁which ▁has ▁been ▁of ▁interest ▁in ▁met am aterial s ▁and ▁in ▁many ▁other ▁fields . ▁ ▁Some ▁useful ▁relations ▁in ▁the ▁calculus ▁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