▁have ▁found ▁many ▁applications , ▁e . g ., ▁the ▁potential ▁theory ▁of ▁edges . ▁ ▁Basic ▁definition ▁ ▁The ▁par abol ic ▁cyl ind rical ▁coordinates ▁ ▁are ▁defined ▁in ▁terms ▁of ▁the ▁Cart esian ▁coordinates ▁ ▁by : ▁ ▁The ▁surfaces ▁of ▁constant ▁ ▁form ▁conf ocal ▁par abol ic ▁cyl ind ers ▁ ▁that ▁open ▁towards ▁, ▁whereas ▁the ▁surfaces ▁of ▁constant ▁ ▁form ▁conf ocal ▁par abol ic ▁cyl ind ers ▁ ▁that ▁open ▁in ▁the ▁opposite ▁direction , ▁i . e ., ▁towards ▁. ▁ ▁The ▁fo ci ▁of ▁all ▁these ▁par abol ic ▁cyl ind ers ▁are ▁located ▁along ▁the ▁line ▁defined ▁by ▁. ▁ ▁The ▁radius ▁ ▁has ▁a ▁simple ▁formula ▁as ▁well ▁ ▁that ▁proves ▁useful ▁in ▁solving ▁the ▁Hamilton – Jac obi ▁equation ▁in ▁par abol ic ▁coordinates ▁for ▁the ▁inverse - square ▁central ▁force ▁problem ▁of ▁mechan ics ; ▁for ▁further ▁details , ▁see ▁the ▁Lap lace – Run ge – L enz ▁vector ▁article . ▁ ▁Sc ale ▁factors ▁ ▁The ▁scale ▁factors ▁for ▁the ▁par abol ic ▁cyl ind rical ▁coordinates ▁ ▁and ▁ ▁are : ▁ ▁D iffer ential ▁elements ▁ ▁The ▁inf init es imal ▁element ▁of ▁volume ▁is ▁ ▁The ▁differential ▁disp lacement ▁is ▁given ▁by : ▁ ▁The ▁differential ▁normal ▁area ▁is ▁given ▁by : ▁ ▁Del ▁ ▁Let ▁ ▁be ▁a ▁scalar ▁field . ▁ ▁The ▁gradient ▁is ▁given ▁by ▁ ▁The ▁Lap la cian ▁is ▁given ▁by ▁ ▁Let ▁ ▁be ▁a ▁vector ▁field ▁of