, ▁Sim mond s , ▁Green ▁and ▁Z erna , ▁ ▁Bas ar ▁and ▁We ich ert , ▁and ▁C iar let . ▁ ▁Basic ▁definitions ▁Let ▁the ▁position ▁of ▁a ▁point ▁in ▁space ▁be ▁character ized ▁by ▁three ▁coordinate ▁variables ▁. ▁ ▁The ▁coordinate ▁curve ▁q 1 ▁represents ▁a ▁curve ▁on ▁which ▁q 2 , ▁q 3 ▁are ▁constant . ▁ ▁Let ▁x ▁be ▁the ▁position ▁vector ▁of ▁the ▁point ▁relative ▁to ▁some ▁origin . ▁ ▁Then , ▁assuming ▁that ▁such ▁a ▁mapping ▁and ▁its ▁inverse ▁exist ▁and ▁are ▁continuous , ▁we ▁can ▁write ▁▁ ▁The ▁fields ▁ ψ i ( x ) ▁are ▁called ▁the ▁cur vil ine ar ▁coordinate ▁functions ▁of ▁the ▁cur vil ine ar ▁coordinate ▁system ▁ ψ ( x ) ▁= ▁ ψ − 1 ( x ). ▁ ▁The ▁q i ▁coordinate ▁curves ▁are ▁defined ▁by ▁the ▁one - parameter ▁family ▁of ▁functions ▁given ▁by ▁ ▁with ▁q j , ▁q k ▁fixed . ▁ ▁Tang ent ▁vector ▁to ▁coordinate ▁curves ▁The ▁tang ent ▁vector ▁to ▁the ▁curve ▁x i ▁at ▁the ▁point ▁x i ( α ) ▁( or ▁to ▁the ▁coordinate ▁curve ▁q i ▁at ▁the ▁point ▁x ) ▁is ▁ ▁Grad ient ▁ ▁S cal ar ▁field ▁Let ▁f ( x ) ▁be ▁a ▁scalar ▁field ▁in ▁space . ▁ ▁Then ▁ ▁The ▁gradient ▁of ▁the ▁field ▁f ▁is ▁defined ▁by ▁ ▁where ▁c ▁is ▁an ▁arbitrary ▁constant ▁vector . ▁ ▁If ▁we ▁define ▁the ▁components ▁ci ▁of ▁c ▁are ▁such ▁that ▁