▁the ▁form : ▁ ▁The ▁diver gence ▁is ▁given ▁by ▁ ▁The ▁curl ▁is ▁given ▁by ▁ ▁Other ▁differential ▁operators ▁can ▁be ▁expressed ▁in ▁the ▁coordinates ▁ ▁by ▁subst itut ing ▁the ▁scale ▁factors ▁into ▁the ▁general ▁formula e ▁found ▁in ▁orthogonal ▁coordinates . ▁ ▁Rel ations hip ▁to ▁other ▁coordinate ▁systems ▁ ▁Rel ations hip ▁to ▁cyl ind rical ▁coordinates ▁: ▁ ▁Par abol ic ▁unit ▁vectors ▁expressed ▁in ▁terms ▁of ▁Cart esian ▁unit ▁vectors : ▁ ▁Par abol ic ▁cyl inder ▁harm on ics ▁▁ ▁Since ▁all ▁of ▁the ▁surfaces ▁of ▁constant ▁, ▁ ▁and ▁ ▁are ▁con ico ids , ▁Lap lace ' s ▁equation ▁is ▁separ able ▁in ▁par abol ic ▁cyl ind rical ▁coordinates . ▁Using ▁the ▁technique ▁of ▁the ▁separation ▁of ▁variables , ▁a ▁separated ▁solution ▁to ▁Lap lace ' s ▁equation ▁may ▁be ▁written : ▁ ▁and ▁Lap lace ' s ▁equation , ▁divided ▁by ▁, ▁is ▁written : ▁ ▁Since ▁the ▁ ▁equation ▁is ▁separate ▁from ▁the ▁rest , ▁we ▁may ▁write ▁ ▁where ▁ ▁is ▁constant . ▁ ▁has ▁the ▁solution : ▁ ▁Sub st itut ing ▁ ▁for ▁, ▁Lap lace ' s ▁equation ▁may ▁now ▁be ▁written : ▁ ▁We ▁may ▁now ▁separate ▁the ▁ ▁and ▁ ▁functions ▁and ▁introduce ▁another ▁constant ▁ ▁to ▁obtain : ▁ ▁The ▁solutions ▁to ▁these ▁equations ▁are ▁the ▁par abol ic ▁cyl inder ▁functions ▁ ▁The ▁par abol ic ▁cyl inder ▁harm on ics ▁for ▁ ▁are ▁now ▁the ▁product ▁of ▁the ▁solutions . ▁The ▁combination